| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 51.4-a1 |
51.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{6} \cdot 17 \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$8.607927960$ |
1.193704832 |
\( -\frac{175090819}{12393} a - \frac{76304683}{4131} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 2 a - 4\) , \( -9 a + 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(2a-4\right){x}-9a+21$ |
| 51.4-a2 |
51.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{2} \cdot 17^{3} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$8.607927960$ |
1.193704832 |
\( \frac{15046475}{44217} a + \frac{7471892}{14739} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(a+2\right){x}$ |
| 51.4-a3 |
51.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3 \cdot 17^{6} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$8.607927960$ |
1.193704832 |
\( -\frac{254055727225}{72412707} a + \frac{250186566457}{24137569} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -4 a - 8\) , \( -5 a - 10\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-4a-8\right){x}-5a-10$ |
| 51.4-a4 |
51.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{3} \cdot 17^{2} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$8.607927960$ |
1.193704832 |
\( \frac{5794240479899}{7803} a + \frac{2516825229074}{2601} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 57 a - 134\) , \( -302 a + 689\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(57a-134\right){x}-302a+689$ |
| 51.4-b1 |
51.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{4} \cdot 17 \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$17.13887047$ |
1.188366851 |
\( -\frac{4229329}{1377} a - \frac{1931068}{459} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -2 a - 2\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-2a-2\right){x}+a+1$ |
| 51.4-b2 |
51.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3 \cdot 17^{4} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.284717618$ |
1.188366851 |
\( -\frac{18043954868496973}{250563} a + \frac{13850394292914794}{83521} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -17 a - 62\) , \( 117 a + 47\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-17a-62\right){x}+117a+47$ |
| 51.4-b3 |
51.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{2} \cdot 17^{2} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.13887047$ |
1.188366851 |
\( \frac{103644466861}{2601} a + \frac{45450927253}{867} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -27 a - 37\) , \( 93 a + 119\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-27a-37\right){x}+93a+119$ |
| 51.4-b4 |
51.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3 \cdot 17 \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.13887047$ |
1.188366851 |
\( \frac{294054958534941149}{51} a + \frac{127695878711234630}{17} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 108 a - 262\) , \( 61 a - 140\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(108a-262\right){x}+61a-140$ |
| 51.4-c1 |
51.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{10} \cdot 17 \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.810507882$ |
0.389747318 |
\( \frac{986535013952}{1003833} a - \frac{756768661969}{334611} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 5 a + 1\) , \( 2 a - 2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5a+1\right){x}+2a-2$ |
| 51.4-c2 |
51.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
51.4 |
\( 3 \cdot 17 \) |
\( 3^{5} \cdot 17^{2} \) |
$0.86100$ |
$(-a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.810507882$ |
0.389747318 |
\( -\frac{244457294466907054}{70227} a + \frac{187643434102522805}{23409} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 15 a - 84\) , \( 77 a - 275\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(15a-84\right){x}+77a-275$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
