| 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 |
| 28.1-a1 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{48} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 1877 a - 10553\) , \( -96600 a + 543228\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(1877a-10553\right){x}-96600a+543228$ |
| 28.1-a2 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{16} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 7 a - 13\) , \( 48 a - 244\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(7a-13\right){x}+48a-244$ |
| 28.1-a3 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{24} \cdot 7^{6} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -48 a + 297\) , \( -819 a + 4634\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-48a+297\right){x}-819a+4634$ |
| 28.1-a4 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{12} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$4$ |
\( 2 \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 392 a - 2183\) , \( -6987 a + 39306\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(392a-2183\right){x}-6987a+39306$ |
| 28.1-a5 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{14} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 117 a - 633\) , \( 1782 a - 10000\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(117a-633\right){x}+1782a-10000$ |
| 28.1-a6 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{30} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$7.027708105$ |
1.371668170 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 30037 a - 169273\) , \( -6361944 a + 35778044\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(30037a-169273\right){x}-6361944a+35778044$ |
| 28.1-b1 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{48} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{4} \) |
$0.208472088$ |
$7.027708105$ |
5.147181506 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 12624511 a - 70993606\) , \( -55404245819 a + 311564412494\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(12624511a-70993606\right){x}-55404245819a+311564412494$ |
| 28.1-b2 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{16} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1.876248795$ |
$7.027708105$ |
5.147181506 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 38561 a - 216826\) , \( 18951201 a - 106571594\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(38561a-216826\right){x}+18951201a-106571594$ |
| 28.1-b3 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{24} \cdot 7^{6} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$0.625416265$ |
$7.027708105$ |
5.147181506 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -331614 a + 1864844\) , \( -395481484 a + 2223980408\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-331614a+1864844\right){x}-395481484a+2223980408$ |
| 28.1-b4 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{12} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$1.250832530$ |
$7.027708105$ |
5.147181506 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 2629786 a - 14788516\) , \( -4302244244 a + 24193564616\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(2629786a-14788516\right){x}-4302244244a+24193564616$ |
| 28.1-b5 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{14} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$3.752497591$ |
$7.027708105$ |
5.147181506 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 778911 a - 4380166\) , \( 847816571 a - 4767675598\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(778911a-4380166\right){x}+847816571a-4767675598$ |
| 28.1-b6 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{30} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{4} \) |
$0.416944176$ |
$7.027708105$ |
5.147181506 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 202154111 a - 1136808646\) , \( -3540993521979 a + 19912689902158\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(202154111a-1136808646\right){x}-3540993521979a+19912689902158$ |
| 28.1-c1 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{4} \) |
$1$ |
$0.436190660$ |
6.895991662 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 111861 a - 629043\) , \( 46029165 a - 258843883\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(111861a-629043\right){x}+46029165a-258843883$ |
| 28.1-c2 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$35.33144352$ |
6.895991662 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 341 a - 1913\) , \( -16361 a + 91999\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(341a-1913\right){x}-16361a+91999$ |
| 28.1-c3 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$3.925715946$ |
6.895991662 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2939 a + 16532\) , \( 334628 a - 1881779\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2939a+16532\right){x}+334628a-1881779$ |
| 28.1-c4 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$4$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.925715946$ |
6.895991662 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 23301 a - 131028\) , \( 3550524 a - 19966291\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(23301a-131028\right){x}+3550524a-19966291$ |
| 28.1-c5 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$35.33144352$ |
6.895991662 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 6901 a - 38803\) , \( -718339 a + 4039555\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6901a-38803\right){x}-718339a+4039555$ |
| 28.1-c6 |
28.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$0.436190660$ |
6.895991662 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 1791221 a - 10072883\) , \( 2950512493 a - 16592134379\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1791221a-10072883\right){x}+2950512493a-16592134379$ |
| 28.1-d1 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$6.531492681$ |
$0.436190660$ |
1.112126396 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 28.1-d2 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.725721409$ |
$35.33144352$ |
1.112126396 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 28.1-d3 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$2.177164227$ |
$3.925715946$ |
1.112126396 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 28.1-d4 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.354328454$ |
$3.925715946$ |
1.112126396 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 28.1-d5 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.451442818$ |
$35.33144352$ |
1.112126396 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 28.1-d6 |
28.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.10631$ |
$(2,a), (2,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$13.06298536$ |
$0.436190660$ |
1.112126396 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
*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.
