| 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 |
| 63.1-a1 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{8} \cdot 7^{16} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.995440694$ |
1.737783746 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 102 a - 306\) , \( 2604 a - 7161\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(102a-306\right){x}+2604a-7161$ |
| 63.1-a2 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{10} \cdot 7^{2} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.963525558$ |
1.737783746 |
\( \frac{103823}{63} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -3 a + 9\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a+9\right){x}$ |
| 63.1-a3 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{14} \cdot 7^{4} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$7.963525558$ |
1.737783746 |
\( \frac{7189057}{3969} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 12 a - 36\) , \( 12 a - 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(12a-36\right){x}+12a-33$ |
| 63.1-a4 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{22} \cdot 7^{2} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.981762779$ |
1.737783746 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 117 a - 351\) , \( -1080 a + 2970\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(117a-351\right){x}-1080a+2970$ |
| 63.1-a5 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{10} \cdot 7^{8} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$3.981762779$ |
1.737783746 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 147 a - 441\) , \( 1632 a - 4488\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(147a-441\right){x}+1632a-4488$ |
| 63.1-a6 |
63.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.990881389$ |
1.737783746 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 2352 a - 7056\) , \( 102180 a - 280995\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2352a-7056\right){x}+102180a-280995$ |
| 63.1-b1 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{8} \cdot 7^{16} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.995440694$ |
1.737783746 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -102 a - 204\) , \( -2604 a - 4557\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-102a-204\right){x}-2604a-4557$ |
| 63.1-b2 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{10} \cdot 7^{2} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.963525558$ |
1.737783746 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 3 a + 6\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(3a+6\right){x}$ |
| 63.1-b3 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{14} \cdot 7^{4} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$7.963525558$ |
1.737783746 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -12 a - 24\) , \( -12 a - 21\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-12a-24\right){x}-12a-21$ |
| 63.1-b4 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{22} \cdot 7^{2} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.981762779$ |
1.737783746 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -117 a - 234\) , \( 1080 a + 1890\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-117a-234\right){x}+1080a+1890$ |
| 63.1-b5 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{10} \cdot 7^{8} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$3.981762779$ |
1.737783746 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -147 a - 294\) , \( -1632 a - 2856\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-147a-294\right){x}-1632a-2856$ |
| 63.1-b6 |
63.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$1.15367$ |
$(-a+2), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.990881389$ |
1.737783746 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -2352 a - 4704\) , \( -102180 a - 178815\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-2352a-4704\right){x}-102180a-178815$ |
*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.
