| 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 |
| 81.1-a1 |
81.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 2^{12} \cdot 3^{18} \) |
$2.16131$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Nn, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$0.610893365$ |
0.151543992 |
\( -\frac{99897344}{27} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 609 a - 2784\) , \( 16546 a - 75024\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(609a-2784\right){x}+16546a-75024$ |
| 81.1-a2 |
81.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 2^{12} \cdot 3^{42} \) |
$2.16131$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Nn, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$0.610893365$ |
0.151543992 |
\( \frac{24288219136}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -3801 a + 17376\) , \( -37256 a + 168912\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-3801a+17376\right){x}-37256a+168912$ |
| 81.1-b1 |
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 |
\( -35735839572482 a + 161923694525417 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -105705 a - 373248\) , \( 46089086 a + 162746508\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-105705a-373248\right){x}+46089086a+162746508$ |
| 81.1-b2 |
81.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{12} \) |
$2.16131$ |
$(3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$0.636714186$ |
$11.52462575$ |
1.820307151 |
\( -1666 a + 8049 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 197367 a + 696928\) , \( 150097888 a + 530014986\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(197367a+696928\right){x}+150097888a+530014986$ |
| 81.1-b3 |
81.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.16131$ |
$(3)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2^{2} \) |
$0.636714186$ |
$23.04925150$ |
1.820307151 |
\( 4913 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -7425 a - 26208\) , \( 563828 a + 1990956\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7425a-26208\right){x}+563828a+1990956$ |
| 81.1-b4 |
81.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{12} \) |
$2.16131$ |
$(3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$0.636714186$ |
$11.52462575$ |
1.820307151 |
\( 1666 a + 6383 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -2319 a - 8189\) , \( -113648 a - 401306\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-2319a-8189\right){x}-113648a-401306$ |
| 81.1-b5 |
81.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.16131$ |
$(3)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2^{2} \) |
$0.636714186$ |
$23.04925150$ |
1.820307151 |
\( 16974593 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -112230 a - 396288\) , \( 41102447 a + 145138044\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-112230a-396288\right){x}+41102447a+145138044$ |
| 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-c1 |
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 |
\( -35735839572482 a + 161923694525417 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1452 a - 5130\) , \( 74570 a + 263319\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-1452a-5130\right){x}+74570a+263319$ |
| 81.1-c2 |
81.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 2^{12} \cdot 3^{12} \) |
$2.16131$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \) |
$1$ |
$11.52462575$ |
2.858907793 |
\( -1666 a + 8049 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 214 a + 760\) , \( 5399 a + 19068\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(214a+760\right){x}+5399a+19068$ |
| 81.1-c3 |
81.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$2.16131$ |
$(3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2^{2} \) |
$1$ |
$23.04925150$ |
2.858907793 |
\( 4913 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -102 a - 360\) , \( 932 a + 3291\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-102a-360\right){x}+932a+3291$ |
| 81.1-c4 |
81.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 2^{12} \cdot 3^{12} \) |
$2.16131$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \) |
$1$ |
$11.52462575$ |
2.858907793 |
\( 1666 a + 6383 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -5 a - 8\) , \( -7 a - 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a-8\right){x}-7a-4$ |
| 81.1-c5 |
81.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$2.16131$ |
$(3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2^{2} \) |
$1$ |
$23.04925150$ |
2.858907793 |
\( 16974593 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1542 a - 5445\) , \( 66560 a + 235032\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-1542a-5445\right){x}+66560a+235032$ |
| 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 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -24672 a - 87120\) , \( 4241462 a + 14977149\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-24672a-87120\right){x}+4241462a+14977149$ |
| 81.1-d1 |
81.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$2.16131$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Nn, 5B.4.1 |
$81$ |
\( 2 \) |
$1$ |
$0.610893365$ |
12.27506342 |
\( -\frac{99897344}{27} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 2784 a - 12615\) , \( 160580 a - 727609\bigr] \) |
${y}^2+{y}={x}^{3}+\left(2784a-12615\right){x}+160580a-727609$ |
| 81.1-d2 |
81.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{42} \) |
$2.16131$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Nn, 5B.4.2 |
$81$ |
\( 2 \) |
$1$ |
$0.610893365$ |
12.27506342 |
\( \frac{24288219136}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -17376 a + 78735\) , \( -361564 a + 1638293\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-17376a+78735\right){x}-361564a+1638293$ |
*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.
