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 |
15.1-a2 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 174 a + 260\) , \( 145 a + 374\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(174a+260\right){x}+145a+374$ |
15.1-b2 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 275 a - 734\) , \( 3747 a - 10492\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(275a-734\right){x}+3747a-10492$ |
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-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 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 273 a - 279\) , \( -1724 a + 5275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(273a-279\right){x}-1724a+5275$ |
1875.1-k2 |
1875.1-k |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{16} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.880738313$ |
$1.631232646$ |
5.525614808 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 4359 a + 6472\) , \( 39901 a + 90526\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4359a+6472\right){x}+39901a+90526$ |
1875.1-bj2 |
1875.1-bj |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{16} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.244129907$ |
1.704752426 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 6872 a - 18375\) , \( 456884 a - 1277134\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(6872a-18375\right){x}+456884a-1277134$ |
*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.