| 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 |
| 2.1-a1 |
2.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{27} \) |
$1.90392$ |
$(2,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$2.780517005$ |
1.396739929 |
\( -\frac{875525544051}{134217728} a - \frac{4425696026269}{134217728} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -2106 a - 17838\) , \( -186745 a - 1579548\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2106a-17838\right){x}-186745a-1579548$ |
| 2.1-a2 |
2.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{3} \) |
$1.90392$ |
$(2,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$25.02465304$ |
1.396739929 |
\( -\frac{1323719786216531}{8} a + \frac{12520054709836611}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -851 a - 7223\) , \( 49834 a + 421493\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-851a-7223\right){x}+49834a+421493$ |
| 2.1-a3 |
2.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{9} \) |
$1.90392$ |
$(2,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 1 \) |
$1$ |
$25.02465304$ |
1.396739929 |
\( -\frac{4098699}{512} a + \frac{39265019}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 109 a + 897\) , \( -27 a - 243\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(109a+897\right){x}-27a-243$ |
| 2.1-b1 |
2.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{27} \) |
$1.90392$ |
$(2,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$0.666354607$ |
$16.10824201$ |
3.594614197 |
\( -\frac{875525544051}{134217728} a - \frac{4425696026269}{134217728} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 100 a - 524\) , \( -849 a + 9727\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(100a-524\right){x}-849a+9727$ |
| 2.1-b2 |
2.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{3} \) |
$1.90392$ |
$(2,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$5.997191470$ |
$1.789804668$ |
3.594614197 |
\( -\frac{1323719786216531}{8} a + \frac{12520054709836611}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 995 a - 8989\) , \( 50909 a - 479811\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(995a-8989\right){x}+50909a-479811$ |
| 2.1-b3 |
2.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{9} \) |
$1.90392$ |
$(2,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1.999063823$ |
$16.10824201$ |
3.594614197 |
\( -\frac{4098699}{512} a + \frac{39265019}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 35 a + 91\) , \( 180 a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(35a+91\right){x}+180a-3$ |
| 2.2-a1 |
2.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{27} \) |
$1.90392$ |
$(2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$2.780517005$ |
1.396739929 |
\( \frac{875525544051}{134217728} a - \frac{331326348145}{8388608} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 2146 a - 19943\) , \( 168907 a - 1596253\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2146a-19943\right){x}+168907a-1596253$ |
| 2.2-a2 |
2.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{9} \) |
$1.90392$ |
$(2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 1 \) |
$1$ |
$25.02465304$ |
1.396739929 |
\( \frac{4098699}{512} a + \frac{2197895}{32} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -69 a + 1007\) , \( 924 a - 7430\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-69a+1007\right){x}+924a-7430$ |
| 2.2-a3 |
2.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{3} \) |
$1.90392$ |
$(2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$25.02465304$ |
1.396739929 |
\( \frac{1323719786216531}{8} a + 1399541865452510 \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 891 a - 8073\) , \( -57057 a + 540967\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(891a-8073\right){x}-57057a+540967$ |
| 2.2-b1 |
2.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{27} \) |
$1.90392$ |
$(2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$0.666354607$ |
$16.10824201$ |
3.594614197 |
\( \frac{875525544051}{134217728} a - \frac{331326348145}{8388608} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -60 a - 503\) , \( 285 a + 2438\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-60a-503\right){x}+285a+2438$ |
| 2.2-b2 |
2.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{9} \) |
$1.90392$ |
$(2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1.999063823$ |
$16.10824201$ |
3.594614197 |
\( \frac{4098699}{512} a + \frac{2197895}{32} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 5 a + 47\) , \( -129 a - 1063\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+47\right){x}-129a-1063$ |
| 2.2-b3 |
2.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{3} \) |
$1.90392$ |
$(2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$5.997191470$ |
$1.789804668$ |
3.594614197 |
\( \frac{1323719786216531}{8} a + 1399541865452510 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -955 a - 8073\) , \( -59938 a - 506942\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-955a-8073\right){x}-59938a-506942$ |
| 4.1-a1 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$0.775241574$ |
$9.925798644$ |
2.576921860 |
\( -\frac{193138614768715}{134217728} a - \frac{102100732476999}{8388608} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -2101 a - 17767\) , \( 156670 a + 1325138\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2101a-17767\right){x}+156670a+1325138$ |
| 4.1-a2 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$2.325724722$ |
$9.925798644$ |
2.576921860 |
\( -\frac{1331}{512} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -6 a - 47\) , \( 541 a + 4562\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-6a-47\right){x}+541a+4562$ |
| 4.1-a3 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$6.977174166$ |
$1.102866516$ |
2.576921860 |
\( \frac{193138614768715}{134217728} a - \frac{1826750334400699}{134217728} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 49 a + 418\) , \( -14617 a - 123648\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(49a+418\right){x}-14617a-123648$ |
| 4.1-b1 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$6.977174166$ |
$1.102866516$ |
2.576921860 |
\( -\frac{193138614768715}{134217728} a - \frac{102100732476999}{8388608} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -388378200 a - 3284994637\) , \( -12470168406441 a - 105475632748311\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-388378200a-3284994637\right){x}-12470168406441a-105475632748311$ |
| 4.1-b2 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$2.325724722$ |
$9.925798644$ |
2.576921860 |
\( -\frac{1331}{512} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -1016945 a - 8601557\) , \( -43108742713 a - 364623938239\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-1016945a-8601557\right){x}-43108742713a-364623938239$ |
| 4.1-b3 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.26415$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$0.775241574$ |
$9.925798644$ |
2.576921860 |
\( \frac{193138614768715}{134217728} a - \frac{1826750334400699}{134217728} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 9150350 a + 77395828\) , \( 1162404214216 a + 9831889675216\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(9150350a+77395828\right){x}+1162404214216a+9831889675216$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$2.39405$ |
$(5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.60621782$ |
1.295591817 |
\( -\frac{47632}{15625} a + \frac{434517}{15625} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 30\) , \( -6 a - 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+30{x}-6a-5$ |
| 5.1-b1 |
5.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$2.39405$ |
$(5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.193444082$ |
$13.55047959$ |
1.755653660 |
\( -\frac{47632}{15625} a + \frac{434517}{15625} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -71 a - 577\) , \( 40575 a + 343196\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-71a-577\right){x}+40575a+343196$ |
| 5.2-a1 |
5.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{6} \) |
$2.39405$ |
$(5,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.60621782$ |
1.295591817 |
\( \frac{47632}{15625} a + \frac{77377}{3125} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( a + 30\) , \( 6 a + 19\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+30\right){x}+6a+19$ |
| 5.2-b1 |
5.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{321}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{6} \) |
$2.39405$ |
$(5,a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.193444082$ |
$13.55047959$ |
1.755653660 |
\( \frac{47632}{15625} a + \frac{77377}{3125} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 72 a - 648\) , \( -40505 a + 383124\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(72a-648\right){x}-40505a+383124$ |
*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.
