| 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 |
| 1.1-a6 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( - 2^{12} \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$16$ |
\( 1 \) |
$1$ |
$2.462613440$ |
1.221798417 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -199511 a - 704499\) , \( -98038287 a - 346185826\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-199511a-704499\right){x}-98038287a-346185826$ |
| 1.1-b6 |
1.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$2.462613440$ |
0.305449604 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2742 a - 9675\) , \( -160318 a - 566108\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2742a-9675\right){x}-160318a-566108$ |
| 16.4-a6 |
16.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{24} \) |
$1.44087$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$3.911869447$ |
$3.262324608$ |
1.582904983 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2009 a - 9109\) , \( -115884 a + 525075\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2009a-9109\right){x}-115884a+525075$ |
| 16.4-b6 |
16.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.44087$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$5.984444571$ |
$3.262324608$ |
2.421555028 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -775 a - 2723\) , \( -23659 a - 83534\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-775a-2723\right){x}-23659a-83534$ |
| 16.5-a6 |
16.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{24} \) |
$1.44087$ |
$(2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \) |
$3.911869447$ |
$6.524649217$ |
1.582904983 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -159 a - 640\) , \( 2144 a + 7296\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-159a-640\right){x}+2144a+7296$ |
| 16.5-b6 |
16.5-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.44087$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \) |
$1.496111142$ |
$6.524649217$ |
2.421555028 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 45 a - 208\) , \( 421 a - 1907\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(45a-208\right){x}+421a-1907$ |
| 81.1-b6 |
81.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 2^{12} \cdot 3^{12} \) |
$2.16131$ |
$(3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$0.636714186$ |
$11.52462575$ |
1.820307151 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -1795635 a - 6340608\) , \( 2629919444 a + 9286584492\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1795635a-6340608\right){x}+2629919444a+9286584492$ |
| 81.1-c6 |
81.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{12} \) |
$2.16131$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \) |
$1$ |
$11.52462575$ |
2.858907793 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -24672 a - 87120\) , \( 4241462 a + 14977149\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-24672a-87120\right){x}+4241462a+14977149$ |
*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.
