| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$1.24382$ |
$(-3a-16)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$25.26639132$ |
2.158653491 |
\( \frac{34633}{32} a + \frac{185257}{32} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 63726 a - 404811\) , \( 473687232 a - 3009027057\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(63726a-404811\right){x}+473687232a-3009027057$ |
| 2.1-b1 |
2.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{3} \) |
$1.24382$ |
$(-3a-16)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.635728358$ |
$23.15587320$ |
2.157350965 |
\( \frac{16857}{8} a + \frac{103481}{8} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -66 a + 413\) , \( -78 a + 499\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-66a+413\right){x}-78a+499$ |
| 2.1-b2 |
2.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$1.24382$ |
$(-3a-16)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$4.907185076$ |
$2.572874800$ |
2.157350965 |
\( \frac{2856208514961}{2} a + \frac{15287427477233}{2} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 984 a - 6257\) , \( 45825 a - 291093\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(984a-6257\right){x}+45825a-291093$ |
| 2.2-a1 |
2.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{5} \) |
$1.24382$ |
$(-3a+19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$25.26639132$ |
2.158653491 |
\( -\frac{34633}{32} a + \frac{109945}{16} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -63728 a - 341084\) , \( -473687233 a - 2535339825\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-63728a-341084\right){x}-473687233a-2535339825$ |
| 2.2-b1 |
2.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( -2 \) |
$1.24382$ |
$(-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$4.907185076$ |
$2.572874800$ |
2.157350965 |
\( -\frac{2856208514961}{2} a + 9071817996097 \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -986 a - 5272\) , \( -45826 a - 245268\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-986a-5272\right){x}-45826a-245268$ |
| 2.2-b2 |
2.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{3} \) |
$1.24382$ |
$(-3a+19)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.635728358$ |
$23.15587320$ |
2.157350965 |
\( -\frac{16857}{8} a + \frac{60169}{4} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 64 a + 348\) , \( 77 a + 421\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(64a+348\right){x}+77a+421$ |
| 4.1-a1 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{12} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.465434457$ |
$16.60067830$ |
2.331136759 |
\( -\frac{22581}{512} a + \frac{67973}{256} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -2 a - 8\) , \( 26 a + 132\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-2a-8\right){x}+26a+132$ |
| 4.1-a2 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.396303373$ |
$1.844519812$ |
2.331136759 |
\( \frac{3012908707275}{134217728} a - \frac{9568548623707}{67108864} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 13 a + 72\) , \( -706 a - 3786\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(13a+72\right){x}-706a-3786$ |
| 4.1-b1 |
4.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.89236563$ |
0.930597599 |
\( -\frac{667625}{16} a + \frac{4237875}{16} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 115 a - 730\) , \( 1669 a - 10602\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(115a-730\right){x}+1669a-10602$ |
| 4.1-b2 |
4.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.89236563$ |
0.930597599 |
\( \frac{667625}{16} a + \frac{1785125}{8} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -115 a - 615\) , \( -1669 a - 8933\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-115a-615\right){x}-1669a-8933$ |
| 4.1-c1 |
4.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.396303373$ |
$1.844519812$ |
2.331136759 |
\( -\frac{3012908707275}{134217728} a - \frac{16124188540139}{134217728} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -14 a + 85\) , \( 705 a - 4492\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-14a+85\right){x}+705a-4492$ |
| 4.1-c2 |
4.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{12} \) |
$1.47916$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.465434457$ |
$16.60067830$ |
2.331136759 |
\( \frac{22581}{512} a + \frac{113365}{512} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( a - 10\) , \( -27 a + 158\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-10\right){x}-27a+158$ |
| 7.1-a1 |
7.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.25900109$ |
1.378036230 |
\( -\frac{10498}{49} a + \frac{68517}{49} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 10\) , \( a\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+10{x}+a$ |
| 7.1-a2 |
7.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( 7 \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$64.51800219$ |
1.378036230 |
\( -\frac{14442}{7} a + \frac{219005}{7} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -7112 a - 38054\) , \( 738021 a + 3950138\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7112a-38054\right){x}+738021a+3950138$ |
| 7.1-b1 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.48467956$ |
0.373454246 |
\( -\frac{10644929118296}{7} a + \frac{67620315009833}{7} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 16020 a + 85756\) , \( -14094595 a - 75439217\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(16020a+85756\right){x}-14094595a-75439217$ |
| 7.1-b2 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{8} \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.742339781$ |
0.373454246 |
\( -\frac{15515624330}{5764801} a + \frac{101748325541}{5764801} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 1754 a - 11142\) , \( 94820 a - 602330\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(1754a-11142\right){x}+94820a-602330$ |
| 7.1-b3 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( 7^{4} \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( \frac{343200}{2401} a + \frac{8797553}{2401} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 264 a - 1677\) , \( -3480 a + 22106\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(264a-1677\right){x}-3480a+22106$ |
| 7.1-b4 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( 7^{2} \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( -\frac{17600880}{49} a + \frac{112558049}{49} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -19300 a - 103289\) , \( -3368242 a - 18028022\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-19300a-103289\right){x}-3368242a-18028022$ |
| 7.1-b5 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( \frac{59639450}{49} a + \frac{319259547}{49} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -54 a - 289\) , \( 500 a + 2676\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-54a-289\right){x}+500a+2676$ |
| 7.1-b6 |
7.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$1.70127$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.48467956$ |
0.373454246 |
\( \frac{126974936}{7} a + \frac{679604487}{7} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 9\) , \( -6\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+9{x}-6$ |
| 7.2-a1 |
7.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{2} \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.25900109$ |
1.378036230 |
\( \frac{10498}{49} a + \frac{58019}{49} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 10\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+10{x}-a+1$ |
| 7.2-a2 |
7.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( 7 \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$64.51800219$ |
1.378036230 |
\( \frac{14442}{7} a + \frac{204563}{7} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 7112 a - 45166\) , \( -738021 a + 4688159\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(7112a-45166\right){x}-738021a+4688159$ |
| 7.2-b1 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( -7 \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.48467956$ |
0.373454246 |
\( -\frac{126974936}{7} a + \frac{806579423}{7} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -a + 9\) , \( -a - 6\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+9\right){x}-a-6$ |
| 7.2-b2 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{2} \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( -\frac{59639450}{49} a + \frac{378898997}{49} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 54 a - 343\) , \( -500 a + 3176\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(54a-343\right){x}-500a+3176$ |
| 7.2-b3 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( 7^{4} \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( -\frac{343200}{2401} a + \frac{9140753}{2401} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -264 a - 1413\) , \( 3480 a + 18626\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-264a-1413\right){x}+3480a+18626$ |
| 7.2-b4 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{8} \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.742339781$ |
0.373454246 |
\( \frac{15515624330}{5764801} a + \frac{86232701211}{5764801} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -1754 a - 9388\) , \( -94820 a - 507510\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-1754a-9388\right){x}-94820a-507510$ |
| 7.2-b5 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( 7^{2} \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$34.96935912$ |
0.373454246 |
\( \frac{17600880}{49} a + \frac{94957169}{49} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 19299 a - 122589\) , \( 3368241 a - 21396264\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(19299a-122589\right){x}+3368241a-21396264$ |
| 7.2-b6 |
7.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( -7 \) |
$1.70127$ |
$(-2a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.48467956$ |
0.373454246 |
\( \frac{10644929118296}{7} a + \frac{56975385891537}{7} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -16021 a + 101776\) , \( 14094594 a - 89533812\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16021a+101776\right){x}+14094594a-89533812$ |
| 8.1-a1 |
8.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{15} \) |
$1.75902$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$0.183274389$ |
$20.32498560$ |
1.909514650 |
\( -\frac{111025}{128} a + \frac{434017}{64} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 946188 a - 6010530\) , \( -1112305897 a + 7065756309\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(946188a-6010530\right){x}-1112305897a+7065756309$ |
| 8.2-a1 |
8.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{15} \) |
$1.75902$ |
$(-3a-16), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$0.183274389$ |
$20.32498560$ |
1.909514650 |
\( \frac{111025}{128} a + \frac{757009}{128} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -946190 a - 5064342\) , \( 1112305896 a + 5953450412\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-946190a-5064342\right){x}+1112305896a+5953450412$ |
| 14.1-a1 |
14.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{13} \cdot 7^{2} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.123183353$ |
$14.30335557$ |
7.827679213 |
\( \frac{88890570788337}{401408} a + \frac{475773766467921}{401408} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -4435214 a + 28174098\) , \( -13150686576 a + 83537763397\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-4435214a+28174098\right){x}-13150686576a+83537763397$ |
| 14.1-b1 |
14.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{8} \cdot 7^{4} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$14.58337219$ |
2.491883142 |
\( -\frac{406244053}{614656} a + \frac{2554488651}{614656} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 3 a - 6\) , \( 7 a - 71\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a-6\right){x}+7a-71$ |
| 14.1-b2 |
14.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7 \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$3.645843049$ |
2.491883142 |
\( -\frac{5487440199154713}{28} a + \frac{34858140504189155}{28} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -34929 a - 186943\) , \( -935876 a - 5009137\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-34929a-186943\right){x}-935876a-5009137$ |
| 14.1-b3 |
14.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$14.58337219$ |
2.491883142 |
\( -\frac{3195028773}{784} a + \frac{20448992267}{784} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -22484 a - 120333\) , \( 4473181 a + 23942029\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-22484a-120333\right){x}+4473181a+23942029$ |
| 14.1-b4 |
14.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.58337219$ |
2.491883142 |
\( \frac{90473019121}{28} a + \frac{484243086125}{28} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 2590 a - 16437\) , \( 192643 a - 1223733\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2590a-16437\right){x}+192643a-1223733$ |
| 14.1-c1 |
14.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2 \cdot 7^{2} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$41.21137123$ |
7.041850119 |
\( \frac{103548353}{98} a - \frac{657598687}{98} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 3715 a - 23590\) , \( -287148 a + 1824073\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3715a-23590\right){x}-287148a+1824073$ |
| 14.1-d1 |
14.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{11} \cdot 7^{4} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.081500795$ |
$15.14189557$ |
0.843474186 |
\( \frac{5383263793}{4917248} a - \frac{14274911855}{4917248} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 567 a - 3480\) , \( -18770 a + 119560\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(567a-3480\right){x}-18770a+119560$ |
| 14.1-e1 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2 \cdot 7^{8} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$18.63371162$ |
$1.704388148$ |
2.713361085 |
\( -\frac{931658058590787}{11529602} a + \frac{5918122789438995}{11529602} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -546 a - 2923\) , \( -18339 a - 98157\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-546a-2923\right){x}-18339a-98157$ |
| 14.1-e2 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.329213953$ |
$13.63510518$ |
2.713361085 |
\( -\frac{124659}{1792} a + \frac{1082349}{1792} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -69 a + 438\) , \( -645 a + 4097\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-69a+438\right){x}-645a+4097$ |
| 14.1-e3 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7 \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.329213953$ |
$54.54042075$ |
2.713361085 |
\( -\frac{10346697}{28} a + \frac{65911347}{28} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 348363 a - 2212924\) , \( -270787959 a + 1720139879\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(348363a-2212924\right){x}-270787959a+1720139879$ |
| 14.1-e4 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.658427906$ |
$27.27021037$ |
2.713361085 |
\( \frac{457083}{784} a + \frac{4739499}{784} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -36 a - 193\) , \( -249 a - 1333\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-36a-193\right){x}-249a-1333$ |
| 14.1-e5 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$9.316855812$ |
$6.817552594$ |
2.713361085 |
\( \frac{66180590487}{9604} a + \frac{354524247387}{9604} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -561 a - 3003\) , \( -17357 a - 92901\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-561a-3003\right){x}-17357a-92901$ |
| 14.1-e6 |
14.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2 \cdot 7^{2} \) |
$2.02316$ |
$(-3a-16), (2a+11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$18.63371162$ |
$1.704388148$ |
2.713361085 |
\( \frac{54509792165758611}{98} a + \frac{291755483665071165}{98} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -8976 a - 48043\) , \( -1117587 a - 5981717\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-8976a-48043\right){x}-1117587a-5981717$ |
| 14.2-a1 |
14.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{14} \cdot 7^{3} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.840963258$ |
$7.479910604$ |
1.612257481 |
\( -\frac{42720626363}{5619712} a + \frac{278778060837}{5619712} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 1413661 a + 7566414\) , \( 1239837576 a + 6636044592\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1413661a+7566414\right){x}+1239837576a+6636044592$ |
| 14.2-a2 |
14.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{7} \cdot 7^{6} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.681926517$ |
$3.739955302$ |
1.612257481 |
\( -\frac{10317171913569809}{15059072} a + \frac{65538320328098063}{15059072} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -6556384 a - 35092056\) , \( 10579415223 a + 56624732594\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6556384a-35092056\right){x}+10579415223a+56624732594$ |
| 14.2-b1 |
14.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{4} \cdot 7^{6} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.154186800$ |
$18.58827758$ |
1.469187796 |
\( \frac{586524413}{1882384} a + \frac{2816274669}{1882384} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 3 a\) , \( -12 a + 92\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+3a{x}-12a+92$ |
| 14.2-b2 |
14.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{8} \cdot 7^{3} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.308373600$ |
$18.58827758$ |
1.469187796 |
\( -\frac{3154031571}{87808} a + \frac{22064896781}{87808} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 4342706 a - 27586369\) , \( -11899750759 a + 75591381219\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4342706a-27586369\right){x}-11899750759a+75591381219$ |
| 14.2-c1 |
14.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7^{3} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.206894820$ |
$45.22754956$ |
4.796712006 |
\( \frac{488375}{686} a - \frac{130963625}{686} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 12 a - 87\) , \( -30 a + 189\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(12a-87\right){x}-30a+189$ |
| 14.2-d1 |
14.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{3} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.475747726$ |
$13.14253506$ |
1.602569367 |
\( -\frac{342463}{1372} a + \frac{875997}{1372} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -a + 7\) , \( -2 a - 19\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+7\right){x}-2a-19$ |
| 14.2-d2 |
14.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{137}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2 \cdot 7^{6} \) |
$2.02316$ |
$(-3a-16), (-2a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.951495452$ |
$6.571267534$ |
1.602569367 |
\( \frac{49822416699}{235298} a + \frac{267021894277}{235298} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -16 a - 73\) , \( -111 a - 603\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16a-73\right){x}-111a-603$ |
*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.
