TITLE: What is the simplest ellipse that goes through exactly 13 lattice points?
QUESTION [22 upvotes]: The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points.

Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$.  What is the simplest ellipse that goes through exactly 13 lattice points?

Here are simple ellipses that go through exactly
5 lattice points through 12 lattice points.
What is an ellipse for 13 points? (Answers added)

$x + 4 x^2 - 8 y + 5 x y + 4 y^2$
$-1 + x^2 - x y + y^2$
$-3 - 13 x + 2 x^2 + x y + 3 y^2$
$-2 - 3 x + x^2 - 2 x y + 2 y^2$
$-7 x + x^2 - x y + y^2$
$-9 x + x^2 + 2 y^2$
$4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$
$-1 - 6 x + x^2 - x y + y^2$
$−9 x^2−11 x y−13 x−4 y^2+17 y−4$
$5 x−10 x^2−6 y−15 x y−6 y^2$
$12+12 x−9 x^2+11 y+5 x y−y^2$
$−10x^2−20 x y−20 x−11 y^2+11 y$
???

There is a Shinzel circle that goes through exactly 13 points, but it is very likely not a minimal solution. For circles, here are the best-known sizes for smallest Lattice Circles that go through a particular number of points. I don't have an answer for 13 points here, either.

REPLY [19 votes]: I will show images first (explanation - after):





























And a few ellipses with 36, 48 lattice points:




We will consider ellipses with format
$a_{11} x^2 + a_{12} xy + a_{22} y^2 + a_{13}x +a_{23} y + a_{33} = 0$,
where 
1) $0 < a_{11} \le a_{22}$ (otherwise we can make exchange $x' = y$, $y'=x$); 
2) $a_{12} \le 0$ (otherwise we can replace $y' = -y$).
We'll denote: 
$a$ $-$ semi-major axis; 
$b$ $-$ semi-minor axis; 
$S$ $-$ square (area); 
$M = \max \{ |a_{11}|, |a_{12}|,|a_{22}|,|a_{13}|,|a_{23}|,|a_{33}| \}$; 
$\Sigma = |a_{11}|+|a_{12}| + |a_{22}| + |a_{13}| + |a_{23}| + |a_{33}|$.
I see 2 (or more) kinds of "smallness": 
A. by semi-major axis ($a$); 
B. by square ($S$); 
(C). (by max coefficient); 
(D). (by sum of coefficients).
A. Priorities to minimize:
 $a \rightarrow S \rightarrow M \rightarrow \Sigma \rightarrow |a_{13}|+|a_{23}| \rightarrow \max\{|a_{13}|,|a_{23}|\}  \rightarrow (a_{13} \le 0)$ ;
B. Priorities to minimize:
 $S \rightarrow a \rightarrow M \rightarrow \Sigma \rightarrow |a_{13}|+|a_{23}| \rightarrow \max\{|a_{13}|,|a_{23}|\}  \rightarrow (a_{13} \le 0)$;
For example, if we will have big amount of ellipses with the same $a$ and $S$, then we will choose ellipse(s) with smallest $M$.
If there are a few ellipses with the same $a$, $S$, $M$, $\Sigma$, then we'll choose ellipse(s) with smallest sum $|a_{13}|+|a_{23}|$.
... while we will get 1 ellipse.
How to calculate $a, S$, if we know ellipse formula? 
We will denote
$$
I_1 = 2a_{11}+ 2a_{22} > 0,
$$
$$
I_2 = \left| \begin{array}{cc} 2a_{11} & a_{12} \\ a_{12} & 2a_{22} \end{array} \right| = 4a_{11}a_{22}-a_{12}^2 > 0,
$$
$$
I_3 = - \left| \begin{array}{ccc} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_{23} & 2a_{33}  \end{array} \right| > 0;
$$
then we can rewrite ellipse equation:
$$
a_{11} x^2 + a_{12} xy + a_{22} y^2  = \frac{I_3}{I_2};
$$
let $\lambda_a, \lambda_b$ are roots of equation
$$
\left| \begin{array}{cc} 2a_{11}-\lambda & a_{12} \\ a_{12} & 2a_{22}-\lambda \end{array} \right| = \lambda^2 - I_1 \lambda + I_2 = 0,
$$
so, 
$\lambda_a = (a_{11}+a_{22})+\sqrt{(a_{22}-a_{11})^2+a_{12}^2}$, 
$\lambda_b = (a_{11}+a_{22})-\sqrt{(a_{22}-a_{11})^2+a_{12}^2}$, 
then 
$\displaystyle a = \frac{\sqrt{I_3 \lambda_a}}{I_2}$,     
$\displaystyle b = \frac{\sqrt{I_3 \lambda_b}}{I_2}$,     
$\displaystyle S = \pi a b = \pi \frac{I_3}{I_2 \sqrt{I_2}}$.

Here is the table of smallest (by $a$ or by $S$) ellipses with $n$
lattice points (up to $n=25$).
\begin{array}{|l|l|l|r|r|}
\hline
\mathrm{Lattice} &  \mathrm{equation}   & \min a / & a & S  \\
\mathrm{points}  &            & \min S   & & \\
\hline  3 & x^2 +xy + y^2 -x-y=0         &    S &  0.81649658 &    1.20919958 \\
\hline  4 & x^2 + y^2 -x-y=0             & a, S &  0.70710678 &    1.57079633 \\
\hline  5 & 2x^2 -xy + 2y^2 -x+y-3=0     & a, S &  1.46059349 &    5.19139671 \\
\hline  6 & x^2 -xy + y^2 -1=0           & a, S &  1.41421356 &    3.62759873 \\
\hline  7 & 2x^2 -xy + 2y^2 -5x-4y-6=0   & a    &  2.92118697 &   20.76558682 \\
          & 2x^2 -xy + 4y^2 -7x-7y-7=0   &    S &  3.09818451 &   20.38568713 \\
\hline  8 & x^2+y^2-x-y-2=0              & a, S &  1.58113883 &    7.85398163 \\
\hline  9 & x^2-xy+3y^2-7x-6y=0          & a, S &  4.81580610 &   38.75014739 \\
\hline 10 & x^2-xy+4y^2-5x-5y-6=0        & a, S &  4.17287179 &   25.95698353 \\
\hline 11 & 3x^2-3xy+4y^2-21x-21y-10=0   & a, S &  8.00878321 &  123.82952163 \\
\hline 12 & x^2+y^2-5x-5y=0              & a    &  3.53553391 &   39.26990817 \\
          & x^2-xy+y^2-2x-2y-3=0         &    S &  3.74165739 &   25.39319110 \\
\hline 13 & 2x^2-xy+3y^2-31x-26y-25=0    & a, S & 11.67004201 &  319.90071698 \\
\hline 14 & x^2-xy+4y^2-12x-9y-13=0      & a, S &  8.34574359 &  103.82793410 \\
\hline 15 & 2x^2-xy+5y^2-39x-36y-41=0    & a, S & 13.28106447 &  340.53118449 \\
\hline 16 & x^2+y^2-7x-7y-8=0            & a, S &  5.70087713 &  102.10176124 \\
\hline 17 & 2x^2-xy+3y^2-51x-51y-54=0    & a, S & 20.21310568 &  959.70215095 \\
\hline 18 & x^2-xy+y^2-7x-7y=0           & a, S &  9.89949494 &  177.75233769 \\
\hline 19 & 2x^2-xy+3y^2-61x-61y-6=0     & a, S & 23.34008401 & 1279.60286793 \\
\hline 20 & 2x^2-xy+2y^2-27x-27y-29=0    & a    & 13.46600658 &  441.26871995 \\
          & x^2+5y^2-31x-25y-12=0        &    S & 16.83745824 &  398.30699525 \\
\hline 21 & 2x^2-xy+5y^2-81x-81y-8=0     & a, S & 26.56212894 & 1362.12473794 \\
\hline 22 & 4x^2-2xy+9y^2-194x-179y-30=0 & a    & 31.84451541 & 2050.29169319 \\
          & x^2-xy+10y^2-73x-61y-24=0    &    S & 40.56562661 & 1609.78378121 \\
\hline 23 & 2x^2-xy+3y^2-105x-105y-54=0  & a    & 40.42621136 & 3838.80860378 \\ 
          & 3x^2-3xy+4y^2-120x-121y-3=0  &    S & 44.04830766 & 3745.84302935 \\
\hline 24 & x^2+y^2-17x-17y-18=0         & a    & 12.74754878 &  510.50880621 \\
          & x^2-xy+y^2-9x-9y-10=0        &    S & 13.49073756 &  330.11148429 \\
\hline 25 & 3x^2-xy+8y^2-279x-271y-84=0  & a, S & 57.49719096 & 6287.89822644 \\
\hline
\cdots & \cdots & \cdots & \cdots & \cdots \\
\end{array}