| 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 |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{24} \) |
$2.20701$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.302029864$ |
$7.279010366$ |
3.021228325 |
\( -\frac{2197}{64} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -9 a + 89\) , \( -15 a + 182\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-9a+89\right){x}-15a+182$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{24} \) |
$2.20701$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.302029864$ |
$7.279010366$ |
3.021228325 |
\( -\frac{2197}{64} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 781648 a - 7216163\) , \( 7968868240 a - 73569584162\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(781648a-7216163\right){x}+7968868240a-73569584162$ |
| 5.1-a1 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.971478438$ |
$8.152758559$ |
2.721066011 |
\( \frac{2248091}{15625} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -7876563 a + 72717700\) , \( -118010544899 a + 1089488055560\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-7876563a+72717700\right){x}-118010544899a+1089488055560$ |
| 5.1-a2 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.971478438$ |
$32.61103423$ |
2.721066011 |
\( -\frac{131794516}{25} a + \frac{1216900233}{25} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 35236318640 a - 325306084074\) , \( -10639094615245036 a + 98221447100901019\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(35236318640a-325306084074\right){x}-10639094615245036a+98221447100901019$ |
| 5.1-a3 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{6} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2 \cdot 3 \) |
$0.485739219$ |
$32.61103423$ |
2.721066011 |
\( \frac{1295029}{125} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 6553812 a - 60505320\) , \( -24613029349 a + 227230554832\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6553812a-60505320\right){x}-24613029349a+227230554832$ |
| 5.1-a4 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.971478438$ |
$32.61103423$ |
2.721066011 |
\( \frac{131794516}{25} a + \frac{1085105717}{25} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 1471562 a - 13585623\) , \( 2448803643 a - 22607660368\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1471562a-13585623\right){x}+2448803643a-22607660368$ |
| 5.1-b1 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.971478438$ |
$8.152758559$ |
2.721066011 |
\( \frac{2248091}{15625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -18 a + 224\) , \( -231 a + 2220\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-18a+224\right){x}-231a+2220$ |
| 5.1-b2 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.971478438$ |
$32.61103423$ |
2.721066011 |
\( -\frac{131794516}{25} a + \frac{1216900233}{25} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 36839 a - 340081\) , \( -11256945 a + 103925500\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(36839a-340081\right){x}-11256945a+103925500$ |
| 5.1-b3 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{6} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2 \cdot 3 \) |
$0.485739219$ |
$32.61103423$ |
2.721066011 |
\( \frac{1295029}{125} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -3 a + 84\) , \( 80\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+84\right){x}+80$ |
| 5.1-b4 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$2.33362$ |
$(4a-37)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.971478438$ |
$32.61103423$ |
2.721066011 |
\( \frac{131794516}{25} a + \frac{1085105717}{25} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( a - 14\) , \( 3 a - 49\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(a-14\right){x}+3a-49$ |
| 8.1-a1 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{12} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.46666831$ |
3.313436071 |
\( -\frac{53253}{16} a - \frac{116491}{4} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -777 a - 6168\) , \( 31293 a + 258272\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-777a-6168\right){x}+31293a+258272$ |
| 8.1-a2 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.607407590$ |
3.313436071 |
\( \frac{3197675937939}{4096} a - \frac{7380334431715}{1024} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2928 a + 24332\) , \( 180300 a + 1484916\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2928a+24332\right){x}+180300a+1484916$ |
| 8.1-b1 |
8.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{12} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.670989331$ |
$14.46666831$ |
4.054529489 |
\( -\frac{53253}{16} a - \frac{116491}{4} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 30 a + 12\) , \( 218 a - 872\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(30a+12\right){x}+218a-872$ |
| 8.1-b2 |
8.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$11.01296799$ |
$1.607407590$ |
4.054529489 |
\( \frac{3197675937939}{4096} a - \frac{7380334431715}{1024} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 1215 a - 10928\) , \( 70666 a - 651256\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1215a-10928\right){x}+70666a-651256$ |
| 8.2-a1 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{20} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.607407590$ |
3.313436071 |
\( -\frac{3197675937939}{4096} a - \frac{26323661788921}{4096} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -2928 a + 27260\) , \( -180300 a + 1665216\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-2928a+27260\right){x}-180300a+1665216$ |
| 8.2-a2 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{12} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.46666831$ |
3.313436071 |
\( \frac{53253}{16} a - \frac{519217}{16} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 777 a - 6945\) , \( -31293 a + 289565\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(777a-6945\right){x}-31293a+289565$ |
| 8.2-b1 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{20} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$11.01296799$ |
$1.607407590$ |
4.054529489 |
\( -\frac{3197675937939}{4096} a - \frac{26323661788921}{4096} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -1176 a - 9676\) , \( -80380 a - 661696\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1176a-9676\right){x}-80380a-661696$ |
| 8.2-b2 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{12} \) |
$2.62459$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.670989331$ |
$14.46666831$ |
4.054529489 |
\( \frac{53253}{16} a - \frac{519217}{16} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 9 a + 79\) , \( -177 a - 1455\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(9a+79\right){x}-177a-1455$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.136838533$ |
$25.53513337$ |
2.400920985 |
\( -\frac{620650477}{729} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -295508 a - 2432563\) , \( 256558100 a + 2112018426\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-295508a-2432563\right){x}+256558100a+2112018426$ |
| 9.1-b1 |
9.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{2} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$12.05843199$ |
$9.928609750$ |
3.427672844 |
\( -\frac{2197}{3} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 781662 a - 7216112\) , \( 2039948421 a - 18833056856\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(781662a-7216112\right){x}+2039948421a-18833056856$ |
| 9.1-b2 |
9.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{4} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$6.029215995$ |
$9.928609750$ |
3.427672844 |
\( \frac{16194277}{9} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 15212037 a - 140439132\) , \( 95437463971 a - 881090557584\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15212037a-140439132\right){x}+95437463971a-881090557584$ |
| 9.1-c1 |
9.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{2} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$12.05843199$ |
$9.928609750$ |
3.427672844 |
\( -\frac{2197}{3} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -9 a + 140\) , \( -24 a + 304\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a+140\right){x}-24a+304$ |
| 9.1-c2 |
9.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{4} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$6.029215995$ |
$9.928609750$ |
3.427672844 |
\( \frac{16194277}{9} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 6 a\) , \( 207 a - 1836\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+6a{x}+207a-1836$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{305}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.70302$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.136838533$ |
$25.53513337$ |
2.400920985 |
\( -\frac{620650477}{729} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 43 a - 388\) , \( -199 a + 1880\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(43a-388\right){x}-199a+1880$ |
*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.
