| 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 |
| 3.1-a1 |
3.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$1.15826$ |
$(-2a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.113989148$ |
$20.15038479$ |
1.399304522 |
\( -\frac{253}{27} a - \frac{1163}{27} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 3 a + 16\) , \( a + 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(3a+16\right){x}+a+35$ |
| 3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{3} \) |
$1.15826$ |
$(2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.113989148$ |
$20.15038479$ |
1.399304522 |
\( \frac{253}{27} a - \frac{472}{9} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -4 a + 20\) , \( -a + 36\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+20\right){x}-a+36$ |
| 6.1-a1 |
6.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{4} \) |
$1.37741$ |
$(-7a+38), (-2a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.669484232$ |
$15.57999680$ |
2.118126267 |
\( \frac{14353}{162} a + \frac{63545}{162} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 9\) , \( -2 a - 8\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+9{x}-2a-8$ |
| 6.1-a2 |
6.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{2} \) |
$1.37741$ |
$(-7a+38), (-2a-9)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.338968465$ |
$31.15999360$ |
2.118126267 |
\( -\frac{147157}{36} a + \frac{913873}{36} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -2712803 a - 12002595\) , \( 3673224085 a + 16251918802\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2712803a-12002595\right){x}+3673224085a+16251918802$ |
| 6.1-a3 |
6.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$1.37741$ |
$(-7a+38), (-2a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.669484232$ |
$31.15999360$ |
2.118126267 |
\( \frac{116701}{48} a + \frac{637115}{48} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 4 a - 16\) , \( -a - 10\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(4a-16\right){x}-a-10$ |
| 6.1-a4 |
6.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3 \) |
$1.37741$ |
$(-7a+38), (-2a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.677936931$ |
$15.57999680$ |
2.118126267 |
\( -\frac{1108475903}{6} a + \frac{6046880261}{6} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -39418998 a - 174406545\) , \( 298973036324 a + 1322784942502\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-39418998a-174406545\right){x}+298973036324a+1322784942502$ |
| 6.1-b1 |
6.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{8} \cdot 3^{11} \) |
$1.37741$ |
$(-7a+38), (-2a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$0.011649964$ |
$12.70696556$ |
0.661352895 |
\( -\frac{5834396582701}{45349632} a - \frac{25805666462171}{45349632} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -130820947 a - 578807959\) , \( 1808429993272 a + 8001269927498\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-130820947a-578807959\right){x}+1808429993272a+8001269927498$ |
| 6.2-a1 |
6.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{20} \) |
$1.37741$ |
$(-7a+38), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.348250326$ |
0.441497929 |
\( -\frac{99900218657837}{55788550416} a + \frac{188089293916871}{18596183472} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 331423 a + 1466362\) , \( 418631874 a + 1852206964\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(331423a+1466362\right){x}+418631874a+1852206964$ |
| 6.2-a2 |
6.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{8} \cdot 3^{10} \) |
$1.37741$ |
$(-7a+38), (2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.39300130$ |
0.441497929 |
\( \frac{2521778623}{15116544} a + \frac{24249763763}{5038848} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -173962 a - 769678\) , \( 73569915 a + 325504860\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-173962a-769678\right){x}+73569915a+325504860$ |
| 6.2-a3 |
6.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{16} \cdot 3^{5} \) |
$1.37741$ |
$(-7a+38), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.696500652$ |
0.441497929 |
\( \frac{17612733229}{15925248} a + \frac{29404052297}{5308416} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -3033 a + 16465\) , \( -264986 a + 1437389\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3033a+16465\right){x}-264986a+1437389$ |
| 6.2-a4 |
6.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{5} \) |
$1.37741$ |
$(-7a+38), (2a-11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$34.78600260$ |
0.441497929 |
\( \frac{5388647303}{3888} a + \frac{21790831339}{1296} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 39 a - 200\) , \( -240 a + 1296\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(39a-200\right){x}-240a+1296$ |
| 6.3-a1 |
6.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{16} \cdot 3^{5} \) |
$1.37741$ |
$(-7a-31), (-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.696500652$ |
0.441497929 |
\( -\frac{17612733229}{15925248} a + \frac{13228111265}{1990656} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 3035 a + 13431\) , \( 268020 a + 1185834\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3035a+13431\right){x}+268020a+1185834$ |
| 6.3-a2 |
6.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( 2^{8} \cdot 3^{10} \) |
$1.37741$ |
$(-7a-31), (-2a-9)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.39300130$ |
0.441497929 |
\( -\frac{2521778623}{15116544} a + \frac{9408883739}{1889568} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 173964 a - 943641\) , \( -73395952 a + 398131134\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(173964a-943641\right){x}-73395952a+398131134$ |
| 6.3-a3 |
6.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{20} \) |
$1.37741$ |
$(-7a-31), (-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.348250326$ |
0.441497929 |
\( \frac{99900218657837}{55788550416} a + \frac{58045957886597}{6973568802} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -331421 a + 1797784\) , \( -418963296 a + 2272636622\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-331421a+1797784\right){x}-418963296a+2272636622$ |
| 6.3-a4 |
6.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{5} \) |
$1.37741$ |
$(-7a-31), (-2a-9)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$34.78600260$ |
0.441497929 |
\( -\frac{5388647303}{3888} a + \frac{8845142665}{486} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -37 a - 162\) , \( 202 a + 894\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-37a-162\right){x}+202a+894$ |
| 6.4-a1 |
6.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{4} \) |
$1.37741$ |
$(-7a-31), (2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.669484232$ |
$15.57999680$ |
2.118126267 |
\( -\frac{14353}{162} a + \frac{12983}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -a + 10\) , \( a - 9\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+10\right){x}+a-9$ |
| 6.4-a2 |
6.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$1.37741$ |
$(-7a-31), (2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.669484232$ |
$31.15999360$ |
2.118126267 |
\( -\frac{116701}{48} a + \frac{31409}{2} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -5 a - 11\) , \( -10\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-11\right){x}-10$ |
| 6.4-a3 |
6.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{2} \) |
$1.37741$ |
$(-7a-31), (2a-11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.338968465$ |
$31.15999360$ |
2.118126267 |
\( \frac{147157}{36} a + \frac{63893}{3} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 2712802 a - 14715397\) , \( -3673224086 a + 19925142888\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2712802a-14715397\right){x}-3673224086a+19925142888$ |
| 6.4-a4 |
6.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3 \) |
$1.37741$ |
$(-7a-31), (2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.677936931$ |
$15.57999680$ |
2.118126267 |
\( \frac{1108475903}{6} a + 823067393 \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 39418997 a - 213825542\) , \( -298973036325 a + 1621757978827\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(39418997a-213825542\right){x}-298973036325a+1621757978827$ |
| 6.4-b1 |
6.4-b |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{8} \cdot 3^{11} \) |
$1.37741$ |
$(-7a-31), (2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$0.011649964$ |
$12.70696556$ |
0.661352895 |
\( \frac{5834396582701}{45349632} a - \frac{1318335960203}{1889568} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 130820946 a - 709628905\) , \( -1808429993272 a + 9809699920770\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(130820946a-709628905\right){x}-1808429993272a+9809699920770$ |
| 8.1-a1 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{5} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.479738915$ |
$19.78946457$ |
1.927893866 |
\( \frac{3937}{2} a - 16276 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -347 a - 1522\) , \( -8933 a - 39507\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-347a-1522\right){x}-8933a-39507$ |
| 8.1-a2 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{7} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.159912971$ |
$19.78946457$ |
1.927893866 |
\( \frac{1}{8} a + 1595 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -3 a + 2\) , \( -3 a + 9\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a+2\right){x}-3a+9$ |
| 8.1-b1 |
8.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 5 \) |
$1$ |
$7.604116941$ |
3.860405487 |
\( \frac{12393}{32} a - \frac{8397}{4} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 6314 a + 27955\) , \( -20372430 a - 90136343\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(6314a+27955\right){x}-20372430a-90136343$ |
| 8.2-a1 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{5} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.479738915$ |
$19.78946457$ |
1.927893866 |
\( -\frac{3937}{2} a - \frac{28615}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 348 a - 1845\) , \( 9280 a - 50285\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(348a-1845\right){x}+9280a-50285$ |
| 8.2-a2 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{7} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.159912971$ |
$19.78946457$ |
1.927893866 |
\( -\frac{1}{8} a + \frac{12761}{8} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 4 a + 23\) , \( 6 a + 29\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(4a+23\right){x}+6a+29$ |
| 8.2-b1 |
8.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{9} \) |
$1.48012$ |
$(-7a+38), (-7a-31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 5 \) |
$1$ |
$7.604116941$ |
3.860405487 |
\( -\frac{12393}{32} a - \frac{54783}{32} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6316 a + 34269\) , \( 20372429 a - 110508773\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-6316a+34269\right){x}+20372429a-110508773$ |
| 9.1-a1 |
9.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$7.676752908$ |
0.779456162 |
\( -\frac{360448}{27} a - \frac{524288}{9} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -52326 a + 283839\) , \( 15381327 a - 83434915\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-52326a+283839\right){x}+15381327a-83434915$ |
| 9.1-a2 |
9.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$7.676752908$ |
0.779456162 |
\( \frac{360448}{27} a - \frac{1933312}{27} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 52326 a + 231513\) , \( -15381327 a - 68053588\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(52326a+231513\right){x}-15381327a-68053588$ |
| 9.1-b1 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{27} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$1.874485042$ |
3.425852158 |
\( -\frac{40047441554978}{282429536481} a + \frac{72378147105725}{94143178827} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171898 a + 932448\) , \( -41983520 a + 227736619\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-171898a+932448\right){x}-41983520a+227736619$ |
| 9.1-b2 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{27} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$1.874485042$ |
3.425852158 |
\( \frac{40047441554978}{282429536481} a + \frac{177086999762197}{282429536481} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 171898 a + 760550\) , \( 41983520 a + 185753099\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(171898a+760550\right){x}+41983520a+185753099$ |
| 9.1-b3 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.497940169$ |
3.425852158 |
\( -\frac{3620768008}{531441} a + \frac{20769783481}{531441} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -48292 a - 213665\) , \( 5535380 a + 24490895\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-48292a-213665\right){x}+5535380a+24490895$ |
| 9.1-b4 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.497940169$ |
3.425852158 |
\( \frac{3620768008}{531441} a + \frac{5716338491}{177147} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 48292 a - 261957\) , \( -5535380 a + 30026275\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(48292a-261957\right){x}-5535380a+30026275$ |
| 9.1-b5 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$7.497940169$ |
3.425852158 |
\( -\frac{346997373866}{729} a + \frac{1883315508131}{729} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 804 a - 4362\) , \( 27606 a - 149747\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(804a-4362\right){x}+27606a-149747$ |
| 9.1-b6 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$7.497940169$ |
3.425852158 |
\( \frac{346997373866}{729} a + \frac{512106044755}{243} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -804 a - 3558\) , \( -27606 a - 122141\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-804a-3558\right){x}-27606a-122141$ |
| 9.1-c1 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.67377185$ |
0.795715207 |
\( -\frac{14080444}{9} a + \frac{25459655}{3} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 46 a - 249\) , \( 396 a - 2148\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(46a-249\right){x}+396a-2148$ |
| 9.1-c2 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.67377185$ |
0.795715207 |
\( -\frac{129160769332}{6561} a + \frac{700626880567}{6561} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 120608 a - 654229\) , \( -50544244 a + 274173658\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(120608a-654229\right){x}-50544244a+274173658$ |
| 9.1-c3 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$31.34754371$ |
0.795715207 |
\( -\frac{81928}{81} a + \frac{826801}{81} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 7668 a - 41594\) , \( -758664 a + 4115319\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(7668a-41594\right){x}-758664a+4115319$ |
| 9.1-c4 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$31.34754371$ |
0.795715207 |
\( \frac{81928}{81} a + \frac{248291}{27} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -7668 a - 33926\) , \( 758664 a + 3356655\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-7668a-33926\right){x}+758664a+3356655$ |
| 9.1-c5 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.67377185$ |
0.795715207 |
\( \frac{14080444}{9} a + \frac{62298521}{9} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -46 a - 203\) , \( -396 a - 1752\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-46a-203\right){x}-396a-1752$ |
| 9.1-c6 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.52435$ |
$(-2a-9), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.67377185$ |
0.795715207 |
\( \frac{129160769332}{6561} a + \frac{190488703745}{2187} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -120608 a - 533621\) , \( 50544244 a + 223629414\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-120608a-533621\right){x}+50544244a+223629414$ |
| 9.2-a1 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{3} \) |
$1.52435$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.64636951$ |
1.657368304 |
\( 598 a + 4499 \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 129357 a - 701673\) , \( 14469665 a - 78489658\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(129357a-701673\right){x}+14469665a-78489658$ |
| 9.2-a2 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.52435$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.64636951$ |
1.657368304 |
\( 676 a + 4601 \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a + 8\) , \( a + 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+8\right){x}+a+3$ |
| 9.2-b1 |
9.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.21466129$ |
1.026243941 |
\( 598 a + 4499 \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 5\) , \( -a - 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+5{x}-a-9$ |
| 9.2-b2 |
9.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.52435$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.21466129$ |
1.026243941 |
\( 676 a + 4601 \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -415172 a - 1836890\) , \( 288105428 a + 1274701983\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-415172a-1836890\right){x}+288105428a+1274701983$ |
| 9.2-c1 |
9.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.067206955$ |
$23.31972861$ |
0.636519687 |
\( \frac{253}{27} a - \frac{472}{9} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -539151 a - 2385428\) , \( 15645584026 a + 69222774138\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-539151a-2385428\right){x}+15645584026a+69222774138$ |
| 9.3-a1 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.52435$ |
$(-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.64636951$ |
1.657368304 |
\( -676 a + 5277 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 7\) , \( -a - 4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+7\right){x}-a-4$ |
| 9.3-a2 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{3} \) |
$1.52435$ |
$(-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$32.64636951$ |
1.657368304 |
\( -598 a + 5097 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -129355 a - 572318\) , \( -14340309 a - 63447676\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-129355a-572318\right){x}-14340309a-63447676$ |
| 9.3-b1 |
9.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.52435$ |
$(-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.21466129$ |
1.026243941 |
\( -676 a + 5277 \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 415171 a - 2252061\) , \( -288105429 a + 1562807412\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(415171a-2252061\right){x}-288105429a+1562807412$ |
| 9.3-b2 |
9.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(-2a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.21466129$ |
1.026243941 |
\( -598 a + 5097 \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -a + 6\) , \( -9\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+6\right){x}-9$ |
| 9.3-c1 |
9.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{97}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.52435$ |
$(-2a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.067206955$ |
$23.31972861$ |
0.636519687 |
\( -\frac{253}{27} a - \frac{1163}{27} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 539150 a - 2924579\) , \( -15645584026 a + 84868358164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(539150a-2924579\right){x}-15645584026a+84868358164$ |
*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.
