| 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 |
| 100.1-a1 |
100.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 5^{2} \) |
$2.60524$ |
$(5,a+2), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.4 |
$1$ |
\( 2 \) |
$0.142831826$ |
$47.01955116$ |
5.827522944 |
\( -\frac{1723025}{4} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -30 a - 129\) , \( 131 a + 536\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-30a-129\right){x}+131a+536$ |
| 100.1-a2 |
100.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{20} \cdot 3^{12} \cdot 5^{10} \) |
$2.60524$ |
$(5,a+2), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.3 |
$1$ |
\( 2 \) |
$3.570795653$ |
$1.880782046$ |
5.827522944 |
\( \frac{1026895}{1024} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 215 a + 921\) , \( -2746 a - 11308\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(215a+921\right){x}-2746a-11308$ |
| 100.1-b1 |
100.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 5^{10} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.20198316$ |
1.323490896 |
\( \frac{185193}{100} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -44 a - 165\) , \( 63 a + 262\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-44a-165\right){x}+63a+262$ |
| 100.1-b2 |
100.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 5^{14} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.050495790$ |
1.323490896 |
\( \frac{154854153}{1250} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -394 a - 1665\) , \( -9397 a - 39008\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-394a-1665\right){x}-9397a-39008$ |
| 100.1-c1 |
100.1-c |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.2, 5B.1.3 |
$25$ |
\( 3 \) |
$1$ |
$0.508604290$ |
4.137441061 |
\( -\frac{349938025}{8} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -126\) , \( -552\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-126{x}-552$ |
| 100.1-c2 |
100.1-c |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{10} \cdot 3^{12} \cdot 5^{4} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.1, 5B.1.4 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$22.88719308$ |
4.137441061 |
\( -\frac{121945}{32} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 38 a - 155\) , \( -295 a + 1579\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(38a-155\right){x}-295a+1579$ |
| 100.1-c3 |
100.1-c |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.1, 5B.1.3 |
$25$ |
\( 3 \) |
$1$ |
$4.577438616$ |
4.137441061 |
\( -\frac{25}{2} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( -2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}-2$ |
| 100.1-c4 |
100.1-c |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{30} \cdot 3^{12} \cdot 5^{4} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.2, 5B.1.4 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$2.543021453$ |
4.137441061 |
\( \frac{46969655}{32768} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -237 a + 1270\) , \( 2250 a - 11351\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-237a+1270\right){x}+2250a-11351$ |
| 100.1-d1 |
100.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{10} \cdot 3^{12} \cdot 5^{20} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 2 \) |
$1$ |
$0.866930840$ |
1.692573337 |
\( \frac{2336752783}{2500000} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 968 a + 4146\) , \( -31133 a - 128499\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(968a+4146\right){x}-31133a-128499$ |
| 100.1-e1 |
100.1-e |
$2$ |
$13$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 5^{32} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$13$ |
13B.5.2 |
$49$ |
\( 2 \) |
$1$ |
$0.839351161$ |
8.921960754 |
\( -\frac{45145776875761017}{2441406250} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -259639 a - 1112715\) , \( 158550493 a + 655237687\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-259639a-1112715\right){x}+158550493a+655237687$ |
| 100.1-e2 |
100.1-e |
$2$ |
$13$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{26} \cdot 3^{12} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$13$ |
13B.5.1 |
$49$ |
\( 2 \) |
$1$ |
$0.839351161$ |
8.921960754 |
\( -\frac{60698457}{40960} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -289 a - 1215\) , \( -8687 a - 36023\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-289a-1215\right){x}-8687a-36023$ |
| 100.1-f1 |
100.1-f |
$2$ |
$13$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{32} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$13$ |
13B.5.2 |
$1$ |
\( 2 \) |
$1$ |
$0.839351161$ |
0.182080831 |
\( -\frac{45145776875761017}{2441406250} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -333816 a - 1372354\) , \( 224539452 a + 922809430\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-333816a-1372354\right){x}+224539452a+922809430$ |
| 100.1-f2 |
100.1-f |
$2$ |
$13$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{26} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$13$ |
13B.5.1 |
$1$ |
\( 2 \) |
$1$ |
$0.839351161$ |
0.182080831 |
\( -\frac{60698457}{40960} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -366 a - 1504\) , \( -13188 a - 54200\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-366a-1504\right){x}-13188a-54200$ |
| 100.1-g1 |
100.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{20} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.866930840$ |
0.188063704 |
\( \frac{2336752783}{2500000} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 1247 a + 5132\) , \( -42965 a - 176574\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(1247a+5132\right){x}-42965a-176574$ |
| 100.1-h1 |
100.1-h |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 3^{12} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$0.508604290$ |
0.496492927 |
\( -\frac{349938025}{8} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 1379 a - 7158\) , \( 63176 a - 323309\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1379a-7158\right){x}+63176a-323309$ |
| 100.1-h2 |
100.1-h |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{4} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$22.88719308$ |
0.496492927 |
\( -\frac{121945}{32} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-3{x}+1$ |
| 100.1-h3 |
100.1-h |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 5^{8} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.577438616$ |
0.496492927 |
\( -\frac{25}{2} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 4 a - 33\) , \( 201 a - 1034\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-33\right){x}+201a-1034$ |
| 100.1-h4 |
100.1-h |
$4$ |
$15$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{30} \cdot 5^{4} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$2.543021453$ |
0.496492927 |
\( \frac{46969655}{32768} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 22\) , \( -9\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+22{x}-9$ |
| 100.1-i1 |
100.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 5^{10} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.20198316$ |
1.323490896 |
\( \frac{185193}{100} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -51 a - 209\) , \( -28 a - 115\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-51a-209\right){x}-28a-115$ |
| 100.1-i2 |
100.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{14} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.050495790$ |
1.323490896 |
\( \frac{154854153}{1250} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -501 a - 2059\) , \( -14508 a - 59625\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-501a-2059\right){x}-14508a-59625$ |
| 100.1-j1 |
100.1-j |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 5^{2} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$47.01955116$ |
1.631995641 |
\( -\frac{1723025}{4} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -35 a - 140\) , \( 151 a + 622\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-35a-140\right){x}+151a+622$ |
| 100.1-j2 |
100.1-j |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{20} \cdot 5^{10} \) |
$2.60524$ |
$(5,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$1.880782046$ |
1.631995641 |
\( \frac{1026895}{1024} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 280 a + 1155\) , \( -3615 a - 14855\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(280a+1155\right){x}-3615a-14855$ |
*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.
