| 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 |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$1.94138$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.437527750$ |
$8.258367178$ |
2.620702670 |
\( 5248 a - 40272 \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 14 a - 14\) , \( 99 a - 490\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(14a-14\right){x}+99a-490$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$1.94138$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.437527750$ |
$8.258367178$ |
2.620702670 |
\( -5248 a - 40272 \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -16 a - 14\) , \( -100 a - 490\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16a-14\right){x}-100a-490$ |
| 4.1-c1 |
4.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$1.94138$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.157573144$ |
$46.13560054$ |
2.839315346 |
\( 5248 a - 40272 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 34\) , \( 20\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-34\right){x}+20$ |
| 4.1-d1 |
4.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$1.94138$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.157573144$ |
$46.13560054$ |
2.839315346 |
\( -5248 a - 40272 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6 a - 34\) , \( -a + 20\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-6a-34\right){x}-a+20$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.37769$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.53621206$ |
1.501886885 |
\( \frac{507397}{9} a - 435898 \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 284 a - 2172\) , \( 8125 a - 62438\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(284a-2172\right){x}+8125a-62438$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.37769$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$19.30300399$ |
2.513037069 |
\( \frac{507397}{9} a - 435898 \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 294 a - 2109\) , \( -6678 a + 51659\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(294a-2109\right){x}-6678a+51659$ |
| 9.1-c1 |
9.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.37769$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.53621206$ |
1.501886885 |
\( -\frac{507397}{9} a - 435898 \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -285 a - 2172\) , \( -8126 a - 62438\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-285a-2172\right){x}-8126a-62438$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.37769$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$19.30300399$ |
2.513037069 |
\( -\frac{507397}{9} a - 435898 \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -295 a - 2109\) , \( 6678 a + 51659\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-295a-2109\right){x}+6678a+51659$ |
| 10.1-a1 |
10.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{6} \) |
$2.44115$ |
$(-3a-23), (a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.709937140$ |
1.486740996 |
\( -\frac{296905503}{4000000} a - \frac{1140357983}{2000000} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -3 a + 3\) , \( 10 a + 102\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+3\right){x}+10a+102$ |
| 10.1-b1 |
10.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{6} \) |
$2.44115$ |
$(-3a-23), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$0.270925733$ |
$4.195444483$ |
4.735351361 |
\( -\frac{296905503}{4000000} a - \frac{1140357983}{2000000} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 7 a + 66\) , \( 9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(7a+66\right){x}+9$ |
| 10.2-a1 |
10.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{6} \) |
$2.44115$ |
$(-3a-23), (-a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.709937140$ |
1.486740996 |
\( \frac{296905503}{4000000} a - \frac{1140357983}{2000000} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 3 a + 3\) , \( -10 a + 102\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+3\right){x}-10a+102$ |
| 10.2-b1 |
10.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{6} \) |
$2.44115$ |
$(-3a-23), (-a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$0.270925733$ |
$4.195444483$ |
4.735351361 |
\( \frac{296905503}{4000000} a - \frac{1140357983}{2000000} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -7 a + 66\) , \( 9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a+66\right){x}+9$ |
| 16.1-a1 |
16.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{36} \) |
$2.74552$ |
$(-3a-23)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$4$ |
\( 2^{2} \) |
$1$ |
$3.469170129$ |
3.613179849 |
\( \frac{2352637}{4096} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 5 a + 91\) , \( 12 a + 205\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(5a+91\right){x}+12a+205$ |
| 16.1-b1 |
16.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.74552$ |
$(-3a-23)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$9.067498572$ |
1.180487764 |
\( -\frac{42875}{8} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 5 a + 77\) , \( 12 a + 135\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(5a+77\right){x}+12a+135$ |
| 16.1-c1 |
16.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$2.74552$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.646146430$ |
$16.78048000$ |
5.780857274 |
\( -2048 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 48760 a - 374513\) , \( -22180635 a + 170372707\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(48760a-374513\right){x}-22180635a+170372707$ |
| 16.1-d1 |
16.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$2.74552$ |
$(-3a-23)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.646146430$ |
$16.78048000$ |
5.780857274 |
\( -2048 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -48760 a - 374513\) , \( 22180635 a + 170372707\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-48760a-374513\right){x}+22180635a+170372707$ |
| 16.1-e1 |
16.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.74552$ |
$(-3a-23)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$9.067498572$ |
1.180487764 |
\( -\frac{42875}{8} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -5 a + 77\) , \( -12 a + 135\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-5a+77\right){x}-12a+135$ |
| 16.1-f1 |
16.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{36} \) |
$2.74552$ |
$(-3a-23)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$4$ |
\( 2^{2} \) |
$1$ |
$3.469170129$ |
3.613179849 |
\( \frac{2352637}{4096} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -5 a + 91\) , \( -12 a + 205\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-5a+91\right){x}-12a+205$ |
| 17.1-a1 |
17.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$2.78745$ |
$(4a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1.364938078$ |
$8.118599020$ |
5.770693753 |
\( \frac{6990412}{83521} a + \frac{32906313}{83521} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 206 a - 1599\) , \( 15119 a - 116137\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(206a-1599\right){x}+15119a-116137$ |
| 17.1-b1 |
17.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.78745$ |
$(4a-31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.429313464$ |
0.306898012 |
\( -\frac{93143493261}{289} a + \frac{715449305042}{289} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -789 a - 6030\) , \( 12481 a + 95880\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-789a-6030\right){x}+12481a+95880$ |
| 17.1-b2 |
17.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$2.78745$ |
$(4a-31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$18.85862692$ |
0.306898012 |
\( \frac{31996}{17} a + \frac{1454663}{17} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -444 a - 3380\) , \( -12982 a - 99705\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-444a-3380\right){x}-12982a-99705$ |
| 17.1-c1 |
17.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$2.78745$ |
$(4a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.412230402$ |
$14.27514461$ |
3.064463979 |
\( \frac{6990412}{83521} a + \frac{32906313}{83521} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 216 a - 1642\) , \( -14062 a + 108016\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(216a-1642\right){x}-14062a+108016$ |
| 17.1-d1 |
17.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.78745$ |
$(4a-31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$30.67281428$ |
0.998315071 |
\( -\frac{93143493261}{289} a + \frac{715449305042}{289} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -779 a - 5997\) , \( -16399 a - 125966\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-779a-5997\right){x}-16399a-125966$ |
| 17.1-d2 |
17.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$2.78745$ |
$(4a-31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$61.34562856$ |
0.998315071 |
\( \frac{31996}{17} a + \frac{1454663}{17} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -434 a - 3347\) , \( 10789 a + 82869\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-434a-3347\right){x}+10789a+82869$ |
| 17.2-a1 |
17.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{4} \) |
$2.78745$ |
$(4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1.364938078$ |
$8.118599020$ |
5.770693753 |
\( -\frac{6990412}{83521} a + \frac{32906313}{83521} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -207 a - 1599\) , \( -15120 a - 116137\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-207a-1599\right){x}-15120a-116137$ |
| 17.2-b1 |
17.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$2.78745$ |
$(4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$18.85862692$ |
0.306898012 |
\( -\frac{31996}{17} a + \frac{1454663}{17} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 443 a - 3380\) , \( 12982 a - 99705\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(443a-3380\right){x}+12982a-99705$ |
| 17.2-b2 |
17.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$2.78745$ |
$(4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.429313464$ |
0.306898012 |
\( \frac{93143493261}{289} a + \frac{715449305042}{289} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 788 a - 6030\) , \( -12481 a + 95880\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(788a-6030\right){x}-12481a+95880$ |
| 17.2-c1 |
17.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{4} \) |
$2.78745$ |
$(4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.412230402$ |
$14.27514461$ |
3.064463979 |
\( -\frac{6990412}{83521} a + \frac{32906313}{83521} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -217 a - 1642\) , \( 14062 a + 108016\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-217a-1642\right){x}+14062a+108016$ |
| 17.2-d1 |
17.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$2.78745$ |
$(4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$61.34562856$ |
0.998315071 |
\( -\frac{31996}{17} a + \frac{1454663}{17} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 433 a - 3347\) , \( -10790 a + 82869\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(433a-3347\right){x}-10790a+82869$ |
| 17.2-d2 |
17.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$2.78745$ |
$(4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$30.67281428$ |
0.998315071 |
\( \frac{93143493261}{289} a + \frac{715449305042}{289} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 778 a - 5997\) , \( 16398 a - 125966\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(778a-5997\right){x}+16398a-125966$ |
| 20.1-a1 |
20.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.90303$ |
$(-3a-23), (-a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$8.572172464$ |
2.511004045 |
\( -\frac{7827456}{15625} a - \frac{65671168}{15625} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 64308 a - 493934\) , \( 59448010 a - 456628805\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(64308a-493934\right){x}+59448010a-456628805$ |
| 20.1-a2 |
20.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{3} \) |
$2.90303$ |
$(-3a-23), (-a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$1$ |
$17.14434492$ |
2.511004045 |
\( \frac{619520032}{125} a + \frac{4861535696}{125} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -9 a + 35\) , \( -17 a + 112\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a+35\right){x}-17a+112$ |
| 20.1-b1 |
20.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.90303$ |
$(-3a-23), (-a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.429631910$ |
0.627800082 |
\( -\frac{7827456}{15625} a - \frac{65671168}{15625} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 64308 a - 493934\) , \( -59448010 a + 456628805\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(64308a-493934\right){x}-59448010a+456628805$ |
| 20.1-b2 |
20.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{3} \) |
$2.90303$ |
$(-3a-23), (-a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$12.85926382$ |
0.627800082 |
\( \frac{619520032}{125} a + \frac{4861535696}{125} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( a + 15\) , \( -14 a - 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a+15\right){x}-14a-92$ |
| 20.2-a1 |
20.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.90303$ |
$(-3a-23), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$8.572172464$ |
2.511004045 |
\( \frac{7827456}{15625} a - \frac{65671168}{15625} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -64308 a - 493934\) , \( -59448010 a - 456628805\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-64308a-493934\right){x}-59448010a-456628805$ |
| 20.2-a2 |
20.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{3} \) |
$2.90303$ |
$(-3a-23), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$1$ |
$17.14434492$ |
2.511004045 |
\( -\frac{619520032}{125} a + \frac{4861535696}{125} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 7 a + 35\) , \( 16 a + 112\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(7a+35\right){x}+16a+112$ |
| 20.2-b1 |
20.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.90303$ |
$(-3a-23), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.429631910$ |
0.627800082 |
\( \frac{7827456}{15625} a - \frac{65671168}{15625} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -64308 a - 493934\) , \( 59448010 a + 456628805\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-64308a-493934\right){x}+59448010a+456628805$ |
| 20.2-b2 |
20.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{3} \) |
$2.90303$ |
$(-3a-23), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$12.85926382$ |
0.627800082 |
\( -\frac{619520032}{125} a + \frac{4861535696}{125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -3 a + 15\) , \( 13 a - 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a+15\right){x}+13a-92$ |
| 25.2-a1 |
25.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$3.06958$ |
$(a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.948013748$ |
$8.224640586$ |
2.113681680 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 66530 a + 511005\) , \( 588080 a + 4517115\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(66530a+511005\right){x}+588080a+4517115$ |
| 25.2-a2 |
25.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$3.06958$ |
$(a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.974006874$ |
$16.44928117$ |
2.113681680 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 15 a + 114\) , \( 45 a + 337\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(15a+114\right){x}+45a+337$ |
| 25.2-b1 |
25.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$3.06958$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1.035756881$ |
$14.16643009$ |
5.730776867 |
\( 35 a + 270 \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( a + 32\) , \( -2 a - 8\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(a+32\right){x}-2a-8$ |
| 25.2-c1 |
25.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$3.06958$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.792844627$ |
$6.437099655$ |
1.993304142 |
\( 35 a + 270 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 11 a + 105\) , \( 34 a + 274\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(11a+105\right){x}+34a+274$ |
| 25.3-a1 |
25.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{9} \) |
$3.06958$ |
$(-a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.974006874$ |
$16.44928117$ |
2.113681680 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 14 a + 85\) , \( 40 a + 293\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(14a+85\right){x}+40a+293$ |
| 25.3-a2 |
25.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{9} \) |
$3.06958$ |
$(-a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.948013748$ |
$8.224640586$ |
2.113681680 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -66501 a + 511034\) , \( -77075 a + 592715\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-66501a+511034\right){x}-77075a+592715$ |
| 25.3-b1 |
25.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$3.06958$ |
$(-a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1.035756881$ |
$14.16643009$ |
5.730776867 |
\( -35 a + 270 \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -2 a + 32\) , \( a - 8\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2a+32\right){x}+a-8$ |
| 25.3-c1 |
25.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$3.06958$ |
$(-a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.792844627$ |
$6.437099655$ |
1.993304142 |
\( -35 a + 270 \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -12 a + 105\) , \( -34 a + 274\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-12a+105\right){x}-34a+274$ |
| 34.1-a1 |
34.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 17^{4} \) |
$3.31485$ |
$(-3a-23), (4a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$1.130336327$ |
$4.636926955$ |
5.458859559 |
\( \frac{31119535900}{83521} a - \frac{967270550337}{334084} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -135 a - 1011\) , \( 2690 a + 20688\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-135a-1011\right){x}+2690a+20688$ |
| 34.1-b1 |
34.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 17^{2} \) |
$3.31485$ |
$(-3a-23), (4a-31)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.388719168$ |
$8.955435714$ |
3.625656527 |
\( -\frac{2541235}{2312} a - \frac{9788013}{1156} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -3248 a - 24911\) , \( -307993 a - 2365654\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-3248a-24911\right){x}-307993a-2365654$ |
| 34.1-b2 |
34.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{2} \cdot 17^{6} \) |
$3.31485$ |
$(-3a-23), (4a-31)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.388719168$ |
$8.955435714$ |
3.625656527 |
\( \frac{191484807134755}{48275138} a - \frac{735370451075577}{24137569} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 17702 a + 136009\) , \( -417253 a - 3204896\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(17702a+136009\right){x}-417253a-3204896$ |
| 34.1-c1 |
34.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{59}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{4} \cdot 17^{4} \) |
$3.31485$ |
$(-3a-23), (4a-31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{4} \) |
$1$ |
$6.916209833$ |
7.203310610 |
\( \frac{31119535900}{83521} a - \frac{967270550337}{334084} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -125 a - 948\) , \( -3340 a - 25647\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-125a-948\right){x}-3340a-25647$ |
*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.
