| 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 |
| 45.2-a1 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.910459118$ |
1.289767919 |
\( -\frac{721}{75} a - \frac{1}{15} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -a + 3\) , \( 2 a - 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+3\right){x}+2a-6$ |
| 45.2-a2 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.738807389$ |
1.289767919 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 2347 a + 4171\) , \( 20528 a + 36708\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(2347a+4171\right){x}+20528a+36708$ |
| 45.2-a3 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$2.955229559$ |
1.289767919 |
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 239 a - 672\) , \( 3320 a - 9264\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(239a-672\right){x}+3320a-9264$ |
| 45.2-a4 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{10} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$5.910459118$ |
1.289767919 |
\( \frac{169820651}{5625} a + \frac{28920482}{375} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 14 a - 42\) , \( 62 a - 174\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(14a-42\right){x}+62a-174$ |
| 45.2-a5 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$1.477614779$ |
1.289767919 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 224 a - 687\) , \( 3434 a - 9366\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(224a-687\right){x}+3434a-9366$ |
| 45.2-a6 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.955229559$ |
1.289767919 |
\( \frac{11403943879867}{2025} a + \frac{4085550627187}{405} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 29 a - 132\) , \( -136 a + 264\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(29a-132\right){x}-136a+264$ |
| 45.2-b1 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.160486773$ |
$17.96736562$ |
1.258473281 |
\( -\frac{721}{75} a - \frac{1}{15} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 3 a + 6\) , \( 4 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+6\right){x}+4a+7$ |
| 45.2-b2 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.320973547$ |
$4.491841405$ |
1.258473281 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 273 a - 279\) , \( -1724 a + 5275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(273a-279\right){x}-1724a+5275$ |
| 45.2-b3 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.641947094$ |
$8.983682810$ |
1.258473281 |
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -12 a - 69\) , \( -110 a - 59\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12a-69\right){x}-110a-59$ |
| 45.2-b4 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{10} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.320973547$ |
$17.96736562$ |
1.258473281 |
\( \frac{169820651}{5625} a + \frac{28920482}{375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -12 a - 24\) , \( 16 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12a-24\right){x}+16a+31$ |
| 45.2-b5 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.283894188$ |
$2.245920702$ |
1.258473281 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -297 a - 579\) , \( -5300 a - 9353\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-297a-579\right){x}-5300a-9353$ |
| 45.2-b6 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.160486773$ |
$8.983682810$ |
1.258473281 |
\( \frac{11403943879867}{2025} a + \frac{4085550627187}{405} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -252 a - 459\) , \( 2770 a + 4957\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-252a-459\right){x}+2770a+4957$ |
*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.
