| 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 |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.17236$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.334130289$ |
$34.45213108$ |
3.280641815 |
\( \frac{745472}{9} a + \frac{4984832}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -2 a - 3\) , \( 14 a + 97\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a-3\right){x}+14a+97$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$2.17236$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.334130289$ |
$34.45213108$ |
3.280641815 |
\( -\frac{745472}{9} a + \frac{5730304}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 4 a - 6\) , \( -11 a + 105\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-6\right){x}-11a+105$ |
| 23.1-a1 |
23.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( 23 \) |
$2.74665$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.810620784$ |
$20.72123952$ |
5.346137385 |
\( \frac{353}{23} a - \frac{5385}{23} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( a - 7\) , \( 4 a - 30\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(a-7\right){x}+4a-30$ |
| 23.2-a1 |
23.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
23.2 |
\( 23 \) |
\( 23 \) |
$2.74665$ |
$(a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.810620784$ |
$20.72123952$ |
5.346137385 |
\( -\frac{353}{23} a - \frac{5032}{23} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -a - 6\) , \( -4 a - 26\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-a-6\right){x}-4a-26$ |
| 28.1-a1 |
28.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{26} \cdot 7 \) |
$2.88510$ |
$(-a+7), (2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 1 \) |
$1$ |
$3.825880340$ |
4.547566204 |
\( -\frac{13271075}{57344} a - \frac{9792721}{7168} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -80 a + 834\) , \( -4565 a + 35020\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-80a+834\right){x}-4565a+35020$ |
| 28.1-b1 |
28.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{22} \cdot 7 \) |
$2.88510$ |
$(-a+7), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 11 \) |
$1.593357032$ |
$2.231103083$ |
5.572143671 |
\( \frac{2977104679977}{14336} a - \frac{3037719430061}{1792} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 1826 a - 13706\) , \( -106604 a + 801477\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(1826a-13706\right){x}-106604a+801477$ |
| 28.1-c1 |
28.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$2.88510$ |
$(-a+7), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$39.90946153$ |
2.843431400 |
\( \frac{1793}{14} a + \frac{5715}{7} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( a - 8\) , \( -14 a + 105\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-8\right){x}-14a+105$ |
| 28.1-c2 |
28.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.88510$ |
$(-a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.434384614$ |
2.843431400 |
\( \frac{8963323958}{343} a + \frac{467371741451}{2744} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -9 a + 67\) , \( 376 a - 2827\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-9a+67\right){x}+376a-2827$ |
| 28.2-a1 |
28.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{26} \cdot 7 \) |
$2.88510$ |
$(-a-6), (2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 1 \) |
$1$ |
$3.825880340$ |
4.547566204 |
\( \frac{13271075}{57344} a - \frac{13087549}{8192} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 107 a + 703\) , \( 5269 a + 34345\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(107a+703\right){x}+5269a+34345$ |
| 28.2-b1 |
28.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{22} \cdot 7 \) |
$2.88510$ |
$(-a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 11 \) |
$1.593357032$ |
$2.231103083$ |
5.572143671 |
\( -\frac{2977104679977}{14336} a - \frac{3046378680073}{2048} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -1828 a - 11879\) , \( 106603 a + 694874\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1828a-11879\right){x}+106603a+694874$ |
| 28.2-c1 |
28.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$2.88510$ |
$(-a-6), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$39.90946153$ |
2.843431400 |
\( -\frac{1793}{14} a + \frac{1889}{2} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -a - 7\) , \( 14 a + 91\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-7\right){x}+14a+91$ |
| 28.2-c2 |
28.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.88510$ |
$(-a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.434384614$ |
2.843431400 |
\( -\frac{8963323958}{343} a + \frac{11001598635}{56} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9 a + 58\) , \( -376 a - 2451\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(9a+58\right){x}-376a-2451$ |
| 41.1-a1 |
41.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
41.1 |
\( 41 \) |
\( 41 \) |
$3.17371$ |
$(a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$48.93491430$ |
0.871617071 |
\( \frac{16175}{41} a + \frac{196098}{41} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 45\) , \( 4 a + 28\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+45{x}+4a+28$ |
| 41.1-a2 |
41.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
41.1 |
\( 41 \) |
\( - 41^{2} \) |
$3.17371$ |
$(a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$24.46745715$ |
0.871617071 |
\( \frac{5828452565}{1681} a + \frac{37991137291}{1681} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -10 a + 120\) , \( -61 a + 516\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+120\right){x}-61a+516$ |
| 41.2-a1 |
41.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
41.2 |
\( 41 \) |
\( - 41^{2} \) |
$3.17371$ |
$(a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$24.46745715$ |
0.871617071 |
\( -\frac{5828452565}{1681} a + \frac{43819589856}{1681} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 10 a + 110\) , \( 61 a + 455\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(10a+110\right){x}+61a+455$ |
| 41.2-a2 |
41.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
41.2 |
\( 41 \) |
\( 41 \) |
$3.17371$ |
$(a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$48.93491430$ |
0.871617071 |
\( -\frac{16175}{41} a + \frac{212273}{41} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 45\) , \( -4 a + 32\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+45{x}-4a+32$ |
| 49.1-a1 |
49.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$3.31834$ |
$(-a+7), (-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$1$ |
\( 7 \) |
$1$ |
$8.360565152$ |
4.169659223 |
\( -\frac{1126039552}{823543} a - \frac{157843456}{16807} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 5 a - 19\) , \( 23 a - 158\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(5a-19\right){x}+23a-158$ |
| 49.1-a2 |
49.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$3.31834$ |
$(-a+7), (-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$1$ |
\( 7 \) |
$1$ |
$8.360565152$ |
4.169659223 |
\( \frac{1126039552}{823543} a - \frac{8860368896}{823543} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -5 a - 14\) , \( -23 a - 135\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5a-14\right){x}-23a-135$ |
| 49.2-a1 |
49.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{9} \) |
$3.31834$ |
$(-a+7)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
|
\( 2 \) |
$1$ |
$4.478700021$ |
7.204104583 |
\( 8344 a - 59683 \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 2 a - 1\) , \( 8 a - 64\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(2a-1\right){x}+8a-64$ |
| 49.2-a2 |
49.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{9} \) |
$3.31834$ |
$(-a+7)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
|
\( 2 \) |
$1$ |
$2.239350010$ |
7.204104583 |
\( -939130878 a + 7060259597 \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 32 a - 246\) , \( 354 a - 2661\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(32a-246\right){x}+354a-2661$ |
| 49.2-b1 |
49.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{8} \) |
$3.31834$ |
$(-a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 3 \) |
$1$ |
$16.92945379$ |
3.618520923 |
\( 4096 a + 28672 \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 2 a - 16\) , \( 2 a - 29\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(2a-16\right){x}+2a-29$ |
| 49.2-c1 |
49.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{2} \) |
$3.31834$ |
$(-a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$0.967946812$ |
$49.35209857$ |
6.806972580 |
\( 4096 a + 28672 \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 14\) , \( a - 14\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+14{x}+a-14$ |
| 49.2-d1 |
49.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{3} \) |
$3.31834$ |
$(-a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
$1$ |
\( 2 \) |
$2.090579042$ |
$43.94562226$ |
6.545594506 |
\( 8344 a - 59683 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 4 a + 30\) , \( 9 a + 67\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(4a+30\right){x}+9a+67$ |
| 49.2-d2 |
49.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{3} \) |
$3.31834$ |
$(-a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
$1$ |
\( 2 \) |
$4.181158084$ |
$21.97281113$ |
6.545594506 |
\( -939130878 a + 7060259597 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 5\) , \( -44 a - 283\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-5\right){x}-44a-283$ |
| 49.3-a1 |
49.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{9} \) |
$3.31834$ |
$(-a-6)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
|
\( 2 \) |
$1$ |
$4.478700021$ |
7.204104583 |
\( -8344 a - 51339 \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -2 a + 1\) , \( -8 a - 56\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a+1\right){x}-8a-56$ |
| 49.3-a2 |
49.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{9} \) |
$3.31834$ |
$(-a-6)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
|
\( 2 \) |
$1$ |
$2.239350010$ |
7.204104583 |
\( 939130878 a + 6121128719 \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -32 a - 214\) , \( -354 a - 2307\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-32a-214\right){x}-354a-2307$ |
| 49.3-b1 |
49.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{8} \) |
$3.31834$ |
$(-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 3 \) |
$1$ |
$16.92945379$ |
3.618520923 |
\( -4096 a + 32768 \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -2 a - 14\) , \( -3 a - 27\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-2a-14\right){x}-3a-27$ |
| 49.3-c1 |
49.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{2} \) |
$3.31834$ |
$(-a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$0.967946812$ |
$49.35209857$ |
6.806972580 |
\( -4096 a + 32768 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 14\) , \( -2 a - 13\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+14{x}-2a-13$ |
| 49.3-d1 |
49.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{3} \) |
$3.31834$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
$1$ |
\( 2 \) |
$2.090579042$ |
$43.94562226$ |
6.545594506 |
\( -8344 a - 51339 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -6 a + 36\) , \( -10 a + 77\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a+36\right){x}-10a+77$ |
| 49.3-d2 |
49.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{3} \) |
$3.31834$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5S4 |
$1$ |
\( 2 \) |
$4.181158084$ |
$21.97281113$ |
6.545594506 |
\( 939130878 a + 6121128719 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -a - 4\) , \( 43 a - 326\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-4\right){x}+43a-326$ |
| 53.1-a1 |
53.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
53.1 |
\( 53 \) |
\( 53^{2} \) |
$3.38408$ |
$(-2a+13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$31.73453747$ |
0.502442706 |
\( -\frac{65681620992}{2809} a + \frac{493791248384}{2809} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -242 a - 1577\) , \( 3380 a + 22030\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-242a-1577\right){x}+3380a+22030$ |
| 53.1-a2 |
53.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
53.1 |
\( 53 \) |
\( 53^{6} \) |
$3.38408$ |
$(-2a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.526059719$ |
0.502442706 |
\( \frac{10430058444029952}{22164361129} a + \frac{68020560801923072}{22164361129} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 1037 a - 7772\) , \( -5250 a + 39446\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1037a-7772\right){x}-5250a+39446$ |
| 53.2-a1 |
53.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
53.2 |
\( 53 \) |
\( 53^{6} \) |
$3.38408$ |
$(2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.526059719$ |
0.502442706 |
\( -\frac{10430058444029952}{22164361129} a + \frac{78450619245953024}{22164361129} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -1035 a - 6736\) , \( 4214 a + 27460\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1035a-6736\right){x}+4214a+27460$ |
| 53.2-a2 |
53.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
53.2 |
\( 53 \) |
\( 53^{2} \) |
$3.38408$ |
$(2a+11)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$31.73453747$ |
0.502442706 |
\( \frac{65681620992}{2809} a + \frac{428109627392}{2809} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 242 a - 1819\) , \( -3380 a + 25410\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(242a-1819\right){x}-3380a+25410$ |
| 59.1-a1 |
59.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
59.1 |
\( 59 \) |
\( 59^{2} \) |
$3.47604$ |
$(-2a+17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$2.507511579$ |
$6.653087229$ |
4.754370724 |
\( -\frac{32603558338560}{3481} a + \frac{245108499746816}{3481} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -8 a - 50\) , \( 64 a + 385\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-8a-50\right){x}+64a+385$ |
| 59.2-a1 |
59.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
59.2 |
\( 59 \) |
\( 59^{2} \) |
$3.47604$ |
$(2a+15)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$2.507511579$ |
$6.653087229$ |
4.754370724 |
\( \frac{32603558338560}{3481} a + \frac{212504941408256}{3481} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 8 a - 58\) , \( -64 a + 449\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8a-58\right){x}-64a+449$ |
| 63.1-a1 |
63.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$3.53351$ |
$(-a+7), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$16.35837921$ |
1.748229392 |
\( -\frac{251005}{1323} a - \frac{124237}{1323} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -23 a + 186\) , \( 50 a - 385\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-23a+186\right){x}+50a-385$ |
| 63.1-a2 |
63.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{12} \cdot 7 \) |
$3.53351$ |
$(-a+7), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$16.35837921$ |
1.748229392 |
\( \frac{574822817}{1701} a + \frac{11252819432}{5103} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 112 a - 829\) , \( 1267 a - 9534\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(112a-829\right){x}+1267a-9534$ |
| 63.2-a1 |
63.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
63.2 |
\( 3^{2} \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$3.53351$ |
$(-a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$16.35837921$ |
1.748229392 |
\( \frac{251005}{1323} a - \frac{7658}{27} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 23 a + 163\) , \( -50 a - 335\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(23a+163\right){x}-50a-335$ |
| 63.2-a2 |
63.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
63.2 |
\( 3^{2} \cdot 7 \) |
\( 3^{12} \cdot 7 \) |
$3.53351$ |
$(-a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$16.35837921$ |
1.748229392 |
\( -\frac{574822817}{1701} a + \frac{1853898269}{729} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -112 a - 717\) , \( -1267 a - 8267\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-112a-717\right){x}-1267a-8267$ |
| 81.1-a1 |
81.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$3.76264$ |
$(3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$0.926763907$ |
$7.797748459$ |
8.238064790 |
\( \frac{745472}{9} a + \frac{4984832}{9} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -27 a - 177\) , \( -203 a - 1323\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-27a-177\right){x}-203a-1323$ |
| 81.1-b1 |
81.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{197}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$3.76264$ |
$(3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$0.926763907$ |
$7.797748459$ |
8.238064790 |
\( -\frac{745472}{9} a + \frac{5730304}{9} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 27 a - 204\) , \( 203 a - 1526\bigr] \) |
${y}^2+{y}={x}^{3}+\left(27a-204\right){x}+203a-1526$ |
*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.
