| 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 |
| 21.1-a1 |
21.1-a |
$4$ |
$6$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{3} \) |
$18.40682$ |
$(-1/3a^3-1/3a^2+3a+2), (-1/3a^3-1/3a^2+2a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2775.234131$ |
3.285379908 |
\( -\frac{45668663}{343} a^{3} - \frac{10461181}{1029} a^{2} + \frac{362255132}{343} a + \frac{367286758}{1029} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2\) , \( a^{2} - a - 3\) , \( -\frac{1}{3} a^{3} - \frac{4}{3} a^{2} + 1\) , \( \frac{2}{3} a^{3} + \frac{2}{3} a^{2} - 2 a - 2\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-2\right){x}^{2}+\left(-\frac{1}{3}a^{3}-\frac{4}{3}a^{2}+1\right){x}+\frac{2}{3}a^{3}+\frac{2}{3}a^{2}-2a-2$ |
| 21.1-a2 |
21.1-a |
$4$ |
$6$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{6} \) |
$18.40682$ |
$(-1/3a^3-1/3a^2+3a+2), (-1/3a^3-1/3a^2+2a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$693.8085328$ |
3.285379908 |
\( \frac{74079408535072}{1058841} a^{3} + \frac{42308209276609}{352947} a^{2} - \frac{67381326805957}{352947} a - \frac{31682015546507}{1058841} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2\) , \( a^{2} - a - 3\) , \( -\frac{26}{3} a^{3} - \frac{44}{3} a^{2} + 25 a + 1\) , \( 47 a^{3} + 84 a^{2} - 120 a - 36\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-2\right){x}^{2}+\left(-\frac{26}{3}a^{3}-\frac{44}{3}a^{2}+25a+1\right){x}+47a^{3}+84a^{2}-120a-36$ |
| 21.1-a3 |
21.1-a |
$4$ |
$6$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7 \) |
$18.40682$ |
$(-1/3a^3-1/3a^2+3a+2), (-1/3a^3-1/3a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$308.3593479$ |
3.285379908 |
\( -\frac{3858185}{189} a^{3} + \frac{5279251}{63} a^{2} - \frac{2609822}{63} a - \frac{4497131}{189} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{28}{3} a^{3} + \frac{1}{3} a^{2} - 75 a - 21\) , \( 37 a^{3} + 4 a^{2} - 292 a - 103\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(\frac{28}{3}a^{3}+\frac{1}{3}a^{2}-75a-21\right){x}+37a^{3}+4a^{2}-292a-103$ |
| 21.1-a4 |
21.1-a |
$4$ |
$6$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{12} \cdot 7^{2} \) |
$18.40682$ |
$(-1/3a^3-1/3a^2+3a+2), (-1/3a^3-1/3a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$77.08983698$ |
3.285379908 |
\( -\frac{841064654581664}{35721} a^{3} + \frac{1030023415263169}{11907} a^{2} - \frac{511441919731475}{11907} a - \frac{943603335173858}{35721} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -19 a^{3} - 3 a^{2} + 150 a + 54\) , \( 148 a^{3} + 16 a^{2} - 1167 a - 407\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-19a^{3}-3a^{2}+150a+54\right){x}+148a^{3}+16a^{2}-1167a-407$ |
*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.
