| 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 |
| 225.3-a1 |
225.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{14} \cdot 5^{8} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.866387513$ |
2.443674873 |
\( -\frac{3309568}{81} a - \frac{7503872}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( 9 a - 96\) , \( -187 a + 172\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-96\right){x}-187a+172$ |
| 225.3-b1 |
225.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3, 7$ |
3B.1.2, 7Ns.3.1 |
$1$ |
\( 2 \) |
$1$ |
$5.475537501$ |
2.389720482 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a - 3\bigr] \) |
${y}^2+a{y}={x}^{3}-a-3$ |
| 225.3-b2 |
225.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3, 7$ |
3B.1.1, 7Ns.3.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$16.42661250$ |
2.389720482 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -10 a + 26\bigr] \) |
${y}^2+a{y}={x}^{3}-10a+26$ |
| 225.3-c1 |
225.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{14} \cdot 5^{8} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.305063867$ |
1.006012348 |
\( -\frac{3309568}{81} a - \frac{7503872}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( -377 a - 680\) , \( 6586 a + 11791\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-377a-680\right){x}+6586a+11791$ |
| 225.3-d1 |
225.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.729432086$ |
$2.553331196$ |
1.927218749 |
\( -3375 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -3\) , \( 3 a - 11\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}-3{x}+3a-11$ |
| 225.3-d2 |
225.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.247061726$ |
$17.87331837$ |
1.927218749 |
\( -3375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -15 a - 27\) , \( 63 a + 113\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-15a-27\right){x}+63a+113$ |
| 225.3-d3 |
225.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.864716043$ |
$2.553331196$ |
1.927218749 |
\( 16581375 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 15 a - 93\) , \( 117 a - 470\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-93\right){x}+117a-470$ |
| 225.3-d4 |
225.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.58597$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.123530863$ |
$17.87331837$ |
1.927218749 |
\( 16581375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -255 a - 462\) , \( 3441 a + 6170\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-255a-462\right){x}+3441a+6170$ |
*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.
