| 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 |
| 870.3-a1 |
870.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{8} \cdot 3 \cdot 5^{2} \cdot 29^{3} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$8.326556130$ |
1.699651152 |
\( -\frac{28528324387}{14633400} a + \frac{45821742803}{9755600} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -48 a - 116\) , \( 1316 a + 3222\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-48a-116\right){x}+1316a+3222$ |
| 870.3-a2 |
870.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 29^{6} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.163278065$ |
1.699651152 |
\( \frac{1937608934699663}{35689399260} a + \frac{4752265478783797}{35689399260} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -328 a + 802\) , \( 15076 a - 36930\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-328a+802\right){x}+15076a-36930$ |
| 870.3-b1 |
870.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{10} \cdot 3^{3} \cdot 5^{4} \cdot 29 \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.373569236$ |
2.065880810 |
\( -\frac{227117199937}{2610000} a - \frac{360405299797}{1740000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 12 a - 37\) , \( -36 a + 80\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(12a-37\right){x}-36a+80$ |
| 870.3-b2 |
870.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{5} \cdot 3^{6} \cdot 5^{2} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.686784618$ |
2.065880810 |
\( \frac{164321962690019249}{4541400} a + \frac{201252481903271483}{2270700} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -48 a + 3\) , \( -300 a + 256\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-48a+3\right){x}-300a+256$ |
| 870.3-c1 |
870.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2 \cdot 3^{4} \cdot 5^{12} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.370716759$ |
3.357556641 |
\( -\frac{9976528898095403699}{3695800781250} a + \frac{12220760114428753717}{1847900390625} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 529 a + 1284\) , \( 8335 a + 20391\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(529a+1284\right){x}+8335a+20391$ |
| 870.3-c2 |
870.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 29^{4} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$5.482867036$ |
3.357556641 |
\( -\frac{12356293209469}{33153796875} a + \frac{233631992666383}{66307593750} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -166 a - 406\) , \( 845 a + 2069\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-166a-406\right){x}+845a+2069$ |
| 870.3-c3 |
870.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2 \cdot 3 \cdot 5^{3} \cdot 29^{8} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$2.741433518$ |
3.357556641 |
\( \frac{1835649767283126517}{375184809720750} a + \frac{1748385073960583663}{62530801620125} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -1181 a - 2896\) , \( -35445 a - 86821\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-1181a-2896\right){x}-35445a-86821$ |
| 870.3-c4 |
870.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{4} \cdot 3 \cdot 5^{3} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$10.96573407$ |
3.357556641 |
\( \frac{196397811499}{315375} a + \frac{641674879813}{420500} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -146 a - 356\) , \( 1315 a + 3221\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-146a-356\right){x}+1315a+3221$ |
| 870.3-d1 |
870.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 29 \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.057118877$ |
$28.76344611$ |
2.682902870 |
\( \frac{55849957}{13050} a - \frac{194881556}{6525} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 41 a - 91\) , \( -220 a + 547\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(41a-91\right){x}-220a+547$ |
| 870.3-e1 |
870.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2^{7} \cdot 3^{8} \cdot 5^{10} \cdot 29^{3} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.888693144$ |
0.725614914 |
\( -\frac{265311510330708841}{308673281250000} a - \frac{147527726173825693}{38584160156250} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 105 a - 348\) , \( -1596 a + 3570\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(105a-348\right){x}-1596a+3570$ |
| 870.3-f1 |
870.3-f |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 29 \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$2.696663326$ |
2.201816386 |
\( \frac{55849957}{13050} a - \frac{194881556}{6525} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -3 a - 4\) , \( -2 a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-4\right){x}-2a-7$ |
| 870.3-g1 |
870.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{10} \cdot 3^{3} \cdot 5^{4} \cdot 29 \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \cdot 5 \) |
$0.050124918$ |
$10.20334567$ |
6.263858058 |
\( -\frac{227117199937}{2610000} a - \frac{360405299797}{1740000} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -143 a - 348\) , \( 1428 a + 3498\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-143a-348\right){x}+1428a+3498$ |
| 870.3-g2 |
870.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{5} \cdot 3^{6} \cdot 5^{2} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \cdot 5 \) |
$0.100249836$ |
$5.101672837$ |
6.263858058 |
\( \frac{164321962690019249}{4541400} a + \frac{201252481903271483}{2270700} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2283 a - 5588\) , \( 92480 a + 226530\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2283a-5588\right){x}+92480a+226530$ |
| 870.3-h1 |
870.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2 \cdot 3^{4} \cdot 5^{12} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.704200591$ |
$3.010739673$ |
1.731107195 |
\( -\frac{9976528898095403699}{3695800781250} a + \frac{12220760114428753717}{1847900390625} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 271 a - 637\) , \( -3583 a + 8749\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(271a-637\right){x}-3583a+8749$ |
| 870.3-h2 |
870.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 29^{4} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.352100295$ |
$6.021479347$ |
1.731107195 |
\( -\frac{12356293209469}{33153796875} a + \frac{233631992666383}{66307593750} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 16 a - 47\) , \( -44 a + 99\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(16a-47\right){x}-44a+99$ |
| 870.3-h3 |
870.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2 \cdot 3 \cdot 5^{3} \cdot 29^{8} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.704200591$ |
$3.010739673$ |
1.731107195 |
\( \frac{1835649767283126517}{375184809720750} a + \frac{1748385073960583663}{62530801620125} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 81 a - 257\) , \( 791 a - 1791\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(81a-257\right){x}+791a-1791$ |
| 870.3-h4 |
870.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{4} \cdot 3 \cdot 5^{3} \cdot 29^{2} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.704200591$ |
$6.021479347$ |
1.731107195 |
\( \frac{196397811499}{315375} a + \frac{641674879813}{420500} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -4 a + 3\) , \( -8 a + 9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-4a+3\right){x}-8a+9$ |
| 870.3-i1 |
870.3-i |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( 2^{7} \cdot 3^{8} \cdot 5^{10} \cdot 29^{3} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
$0.027894908$ |
$1.005858209$ |
4.811000079 |
\( -\frac{265311510330708841}{308673281250000} a - \frac{147527726173825693}{38584160156250} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -1802 a - 4416\) , \( 73105 a + 179055\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1802a-4416\right){x}+73105a+179055$ |
| 870.3-j1 |
870.3-j |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{8} \cdot 3 \cdot 5^{2} \cdot 29^{3} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.131808377$ |
$6.048215378$ |
3.905493172 |
\( -\frac{28528324387}{14633400} a + \frac{45821742803}{9755600} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2 a - 8\) , \( -4 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2a-8\right){x}-4a-1$ |
| 870.3-j2 |
870.3-j |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
870.3 |
\( 2 \cdot 3 \cdot 5 \cdot 29 \) |
\( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 29^{6} \) |
$2.37752$ |
$(-a+2), (a+3), (-a-1), (3a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.263616754$ |
$3.024107689$ |
3.905493172 |
\( \frac{1937608934699663}{35689399260} a + \frac{4752265478783797}{35689399260} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -18 a - 28\) , \( -84 a - 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-18a-28\right){x}-84a-121$ |
*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.
