| 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-a1 |
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 |
\( -35735839572482 a + 161923694525417 \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -11741 a - 41459\) , \( -1744313 a - 6159394\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-11741a-41459\right){x}-1744313a-6159394$ |
| 1.1-a2 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$39.40181504$ |
1.221798417 |
\( -1666 a + 8049 \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 21930 a + 77442\) , \( -5599614 a - 19772961\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(21930a+77442\right){x}-5599614a-19772961$ |
| 1.1-a3 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 2^{12} \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$39.40181504$ |
1.221798417 |
\( 4913 \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -821 a - 2899\) , \( -23499 a - 82978\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-821a-2899\right){x}-23499a-82978$ |
| 1.1-a4 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$39.40181504$ |
1.221798417 |
\( 1666 a + 6383 \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -258 a - 905\) , \( 4597 a + 16233\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-258a-905\right){x}+4597a+16233$ |
| 1.1-a5 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 2^{12} \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2Cs, 3Nn |
$16$ |
\( 1 \) |
$1$ |
$9.850453760$ |
1.221798417 |
\( 16974593 \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -12466 a - 44019\) , \( -1561926 a - 5515362\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-12466a-44019\right){x}-1561926a-5515362$ |
| 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-b1 |
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 |
\( -35735839572482 a + 161923694525417 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -162 a - 565\) , \( -2952 a - 10428\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-162a-565\right){x}-2952a-10428$ |
| 1.1-b2 |
1.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( - 2^{12} \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$39.40181504$ |
0.305449604 |
\( -1666 a + 8049 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 24 a + 80\) , \( -168 a - 596\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(24a+80\right){x}-168a-596$ |
| 1.1-b3 |
1.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$39.40181504$ |
0.305449604 |
\( 4913 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -12 a - 35\) , \( -48 a - 174\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-12a-35\right){x}-48a-174$ |
| 1.1-b4 |
1.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( - 2^{12} \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$39.40181504$ |
0.305449604 |
\( 1666 a + 6383 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
| 1.1-b5 |
1.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.72044$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$9.850453760$ |
0.305449604 |
\( 16974593 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -172 a - 600\) , \( -2667 a - 9422\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-172a-600\right){x}-2667a-9422$ |
| 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 |
\( 35735839572482 a + 126187854952935 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2742 a - 9675\) , \( -160318 a - 566108\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2742a-9675\right){x}-160318a-566108$ |
| 8.1-a1 |
8.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.890968917$ |
$16.68222306$ |
1.304248605 |
\( \frac{4745}{16} a + 1065 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+4\right){x}-2a+4$ |
| 8.1-a2 |
8.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.836453376$ |
$3.707160680$ |
1.304248605 |
\( -\frac{85199050120395}{64} a + \frac{24127992269285}{4} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 407 a - 1861\) , \( 10141 a - 45965\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(407a-1861\right){x}+10141a-45965$ |
| 8.1-a3 |
8.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.945484458$ |
$33.36444612$ |
1.304248605 |
\( \frac{1144845}{4} a + 1033460 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 2 a - 21\) , \( 23 a - 109\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-21\right){x}+23a-109$ |
| 8.1-a4 |
8.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{20} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$5.672906752$ |
$1.853580340$ |
1.304248605 |
\( \frac{6942650905}{4096} a + \frac{693723785}{256} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a - 116\) , \( 204 a - 940\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a-116\right){x}+204a-940$ |
| 8.1-b1 |
8.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{24} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.68222306$ |
2.069175109 |
\( \frac{4745}{16} a + 1065 \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -2414 a + 10928\) , \( 1263405 a - 5724656\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-2414a+10928\right){x}+1263405a-5724656$ |
| 8.1-b2 |
8.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.707160680$ |
2.069175109 |
\( -\frac{85199050120395}{64} a + \frac{24127992269285}{4} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 1869 a - 8460\) , \( 90200 a - 408703\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(1869a-8460\right){x}+90200a-408703$ |
| 8.1-b3 |
8.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{6} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$33.36444612$ |
2.069175109 |
\( \frac{1144845}{4} a + 1033460 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 24 a - 100\) , \( 143 a - 643\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(24a-100\right){x}+143a-643$ |
| 8.1-b4 |
8.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{32} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.853580340$ |
2.069175109 |
\( \frac{6942650905}{4096} a + \frac{693723785}{256} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 375611 a - 1701952\) , \( 252485597 a - 1144044784\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(375611a-1701952\right){x}+252485597a-1144044784$ |
| 8.2-a1 |
8.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{20} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$5.672906752$ |
$1.853580340$ |
1.304248605 |
\( -\frac{6942650905}{4096} a + \frac{18042231465}{4096} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 94\) , \( -205 a - 736\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24a-94\right){x}-205a-736$ |
| 8.2-a2 |
8.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.890968917$ |
$16.68222306$ |
1.304248605 |
\( -\frac{4745}{16} a + \frac{21785}{16} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+a+2$ |
| 8.2-a3 |
8.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.945484458$ |
$33.36444612$ |
1.304248605 |
\( -\frac{1144845}{4} a + \frac{5278685}{4} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 64 a - 280\) , \( -599 a + 2720\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(64a-280\right){x}-599a+2720$ |
| 8.2-a4 |
8.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.836453376$ |
$3.707160680$ |
1.304248605 |
\( \frac{85199050120395}{64} a + \frac{300848826188165}{64} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 184 a - 840\) , \( 1917 a - 8736\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(184a-840\right){x}+1917a-8736$ |
| 8.2-b1 |
8.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{32} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.853580340$ |
2.069175109 |
\( -\frac{6942650905}{4096} a + \frac{18042231465}{4096} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1867 a - 6592\) , \( -84828 a - 299540\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1867a-6592\right){x}-84828a-299540$ |
| 8.2-b2 |
8.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{24} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.68222306$ |
2.069175109 |
\( -\frac{4745}{16} a + \frac{21785}{16} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 13 a + 48\) , \( -468 a - 1652\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(13a+48\right){x}-468a-1652$ |
| 8.2-b3 |
8.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{6} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$33.36444612$ |
2.069175109 |
\( -\frac{1144845}{4} a + \frac{5278685}{4} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -24 a - 76\) , \( -143 a - 500\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-24a-76\right){x}-143a-500$ |
| 8.2-b4 |
8.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.21162$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.707160680$ |
2.069175109 |
\( \frac{85199050120395}{64} a + \frac{300848826188165}{64} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -1869 a - 6591\) , \( -90200 a - 318503\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-1869a-6591\right){x}-90200a-318503$ |
| 9.1-a1 |
9.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.24783$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Nn, 5B.1.4 |
$1$ |
\( 1 \) |
$1$ |
$42.47381586$ |
5.268228478 |
\( -\frac{99897344}{27} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 68 a - 304\) , \( -693 a + 3136\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(68a-304\right){x}-693a+3136$ |
| 9.1-a2 |
9.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{30} \) |
$1.24783$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Nn, 5B.1.3 |
$25$ |
\( 1 \) |
$1$ |
$1.698952634$ |
5.268228478 |
\( \frac{24288219136}{14348907} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -422 a + 1936\) , \( 1883 a - 8512\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-422a+1936\right){x}+1883a-8512$ |
| 9.1-b1 |
9.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.24783$ |
$(3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Nn, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$42.47381586$ |
0.210729139 |
\( -\frac{99897344}{27} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 309 a - 1396\) , \( -6414 a + 29064\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(309a-1396\right){x}-6414a+29064$ |
| 9.1-b2 |
9.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{30} \) |
$1.24783$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Nn, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.698952634$ |
0.210729139 |
\( \frac{24288219136}{14348907} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -1931 a + 8754\) , \( 16308 a - 73892\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1931a+8754\right){x}+16308a-73892$ |
| 10.1-a1 |
10.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{15} \cdot 5^{3} \) |
$1.28114$ |
$(2,a), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$1$ |
$1.791254889$ |
1.999600423 |
\( -\frac{140100188241}{200} a - \frac{494712157087}{200} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -5519 a - 19488\) , \( -446253 a - 1575780\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-5519a-19488\right){x}-446253a-1575780$ |
| 10.1-a2 |
10.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{13} \cdot 5 \) |
$1.28114$ |
$(2,a), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$16.12129400$ |
1.999600423 |
\( -\frac{2637}{10} a + \frac{541}{10} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -59 a - 208\) , \( -766 a - 2708\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-59a-208\right){x}-766a-2708$ |
| 10.1-b1 |
10.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{3} \) |
$1.28114$ |
$(2,a), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.791254889$ |
0.222177824 |
\( -\frac{140100188241}{200} a - \frac{494712157087}{200} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -72 a - 256\) , \( -998 a - 3528\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-72a-256\right){x}-998a-3528$ |
| 10.1-b2 |
10.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5 \) |
$1.28114$ |
$(2,a), (5,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$16.12129400$ |
0.222177824 |
\( -\frac{2637}{10} a + \frac{541}{10} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 3 a + 9\) , \( 2 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a+9\right){x}+2a+3$ |
| 10.2-a1 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{15} \cdot 5^{3} \) |
$1.28114$ |
$(2,a+1), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$1$ |
$1.791254889$ |
1.999600423 |
\( \frac{140100188241}{200} a - \frac{79351543166}{25} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 13 a - 80\) , \( 107 a - 512\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(13a-80\right){x}+107a-512$ |
| 10.2-a2 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{13} \cdot 5 \) |
$1.28114$ |
$(2,a+1), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$16.12129400$ |
1.999600423 |
\( \frac{2637}{10} a - \frac{1048}{5} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -2 a\) , \( -a\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}-2a{x}-a$ |
| 10.2-b1 |
10.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{3} \) |
$1.28114$ |
$(2,a+1), (5,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.791254889$ |
0.222177824 |
\( \frac{140100188241}{200} a - \frac{79351543166}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -1245 a - 4375\) , \( -68756 a - 242770\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1245a-4375\right){x}-68756a-242770$ |
| 10.2-b2 |
10.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5 \) |
$1.28114$ |
$(2,a+1), (5,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$16.12129400$ |
0.222177824 |
\( \frac{2637}{10} a - \frac{1048}{5} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 120 a + 445\) , \( 1355 a + 4801\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(120a+445\right){x}+1355a+4801$ |
| 14.2-a1 |
14.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{2} \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.167868700$ |
$27.34314790$ |
1.138653427 |
\( \frac{1148017}{3136} a + \frac{144765}{196} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 3 a - 3\) , \( -6 a + 34\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-3\right){x}-6a+34$ |
| 14.2-a2 |
14.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{15} \cdot 7 \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.335737401$ |
$54.68629581$ |
1.138653427 |
\( -\frac{8716175}{56} a + \frac{5279401}{7} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 38 a - 160\) , \( -196 a + 896\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(38a-160\right){x}-196a+896$ |
| 14.2-b1 |
14.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{15} \cdot 7 \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$1.313405734$ |
0.651631726 |
\( -\frac{314247100733331753}{56} a + \frac{177986763968809445}{7} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -5 a - 248\) , \( -411 a - 3168\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-5a-248\right){x}-411a-3168$ |
| 14.2-b2 |
14.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{12} \cdot 7^{4} \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.253622939$ |
0.651631726 |
\( \frac{19207604513}{9834496} a - \frac{4276213743}{614656} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -3 a - 8\) , \( -9 a - 35\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a-8\right){x}-9a-35$ |
| 14.2-b3 |
14.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{2} \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$5.253622939$ |
0.651631726 |
\( -\frac{82417844415}{3136} a + \frac{23500331573}{196} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -45 a - 168\) , \( -451 a - 1632\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-45a-168\right){x}-451a-1632$ |
| 14.2-b4 |
14.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7 \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$5.253622939$ |
0.651631726 |
\( \frac{4618208311865}{56} a + \frac{2038436197001}{7} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -9585 a - 33838\) , \( -1037541 a - 3663697\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-9585a-33838\right){x}-1037541a-3663697$ |
| 14.2-c1 |
14.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{3} \cdot 7 \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 3 \) |
$1$ |
$1.313405734$ |
1.954895180 |
\( -\frac{314247100733331753}{56} a + \frac{177986763968809445}{7} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 133 a - 600\) , \( 1801 a - 8174\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(133a-600\right){x}+1801a-8174$ |
| 14.2-c2 |
14.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{24} \cdot 7^{4} \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$5.253622939$ |
1.954895180 |
\( \frac{19207604513}{9834496} a - \frac{4276213743}{614656} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 1649 a - 7451\) , \( 77153 a - 349562\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1649a-7451\right){x}+77153a-349562$ |
| 14.2-c3 |
14.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{2} \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.253622939$ |
1.954895180 |
\( -\frac{82417844415}{3136} a + \frac{23500331573}{196} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 8 a - 35\) , \( 33 a - 158\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(8a-35\right){x}+33a-158$ |
| 14.2-c4 |
14.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{15} \cdot 7 \) |
$1.39357$ |
$(2,a+1), (7,a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 3 \) |
$1$ |
$5.253622939$ |
1.954895180 |
\( \frac{4618208311865}{56} a + \frac{2038436197001}{7} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( a - 64\) , \( -43 a - 324\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a-64\right){x}-43a-324$ |
*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.
