| 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 |
| 147.1-a1 |
147.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{10} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.919140139$ |
2.548034410 |
\( \frac{1381357}{441} a - \frac{1307989}{147} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 17 a - 46\) , \( 57 a - 159\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a-46\right){x}+57a-159$ |
| 147.1-a2 |
147.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{8} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.919140139$ |
2.548034410 |
\( -\frac{28686222041}{567} a + \frac{80304512846}{567} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 262 a - 781\) , \( 3683 a - 10204\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(262a-781\right){x}+3683a-10204$ |
| 147.1-b1 |
147.1-b |
$2$ |
$13$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{26} \cdot 7^{4} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$13$ |
13B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$3.686606385$ |
1.608966934 |
\( -\frac{1713910976512}{1594323} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -912\) , \( 10919\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-912{x}+10919$ |
| 147.1-b2 |
147.1-b |
$2$ |
$13$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$13$ |
13B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$3.686606385$ |
1.608966934 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-2{x}-1$ |
| 147.1-c1 |
147.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{10} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.919140139$ |
2.548034410 |
\( -\frac{1381357}{441} a - \frac{51890}{9} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -16 a - 30\) , \( -42 a - 72\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-16a-30\right){x}-42a-72$ |
| 147.1-c2 |
147.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{8} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.919140139$ |
2.548034410 |
\( \frac{28686222041}{567} a + \frac{5735365645}{63} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -261 a - 520\) , \( -3423 a - 6001\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-261a-520\right){x}-3423a-6001$ |
| 147.1-d1 |
147.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{10} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.451058583$ |
$4.350866838$ |
1.713006800 |
\( -\frac{1381357}{441} a - \frac{51890}{9} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 10 a - 28\) , \( 91 a - 254\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-28\right){x}+91a-254$ |
| 147.1-d2 |
147.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{8} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.902117166$ |
$4.350866838$ |
1.713006800 |
\( \frac{28686222041}{567} a + \frac{5735365645}{63} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 255 a - 763\) , \( 3423 a - 9466\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(255a-763\right){x}+3423a-9466$ |
| 147.1-e1 |
147.1-e |
$2$ |
$13$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{26} \cdot 7^{4} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$13$ |
13B.3.2 |
$1$ |
\( 2 \cdot 13 \) |
$0.991510447$ |
$0.188631360$ |
2.122290778 |
\( -\frac{1713910976512}{1594323} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 4562 a - 12771\) , \( 257501 a - 718820\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(4562a-12771\right){x}+257501a-718820$ |
| 147.1-e2 |
147.1-e |
$2$ |
$13$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$13$ |
13B.3.1 |
$1$ |
\( 2 \) |
$0.076270034$ |
$31.87869985$ |
2.122290778 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 12 a - 31\) , \( -29 a + 80\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(12a-31\right){x}-29a+80$ |
| 147.1-f1 |
147.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{10} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.451058583$ |
$4.350866838$ |
1.713006800 |
\( \frac{1381357}{441} a - \frac{1307989}{147} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -7 a - 15\) , \( -119 a - 209\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-7a-15\right){x}-119a-209$ |
| 147.1-f2 |
147.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{8} \) |
$1.42586$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.902117166$ |
$4.350866838$ |
1.713006800 |
\( -\frac{28686222041}{567} a + \frac{80304512846}{567} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -252 a - 505\) , \( -4186 a - 7314\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-252a-505\right){x}-4186a-7314$ |
*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.
