| 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 |
| 19.1-a1 |
19.1-a |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$9.24336$ |
$(a^3-2a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$57.02749930$ |
1.593189328 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{2} - a - 4\) , \( a^{3} - 2 a^{2} - 4 a + 8\) , \( -a^{3} + 3 a^{2} + 3 a - 11\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(a^{3}-2a^{2}-4a+8\right){x}-a^{3}+3a^{2}+3a-11$ |
| 19.1-b1 |
19.1-b |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$9.24336$ |
$(a^3-2a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.008997105$ |
$2008.533719$ |
2.019412344 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( -a^{3} + 6 a + 4\) , \( a^{2} - 3\) , \( -7 a^{3} - 4 a^{2} + 32 a + 32\) , \( 11 a^{3} + 9 a^{2} - 43 a - 45\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(-7a^{3}-4a^{2}+32a+32\right){x}+11a^{3}+9a^{2}-43a-45$ |
| 361.3-a1 |
361.3-a |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
361.3 |
\( 19^{2} \) |
\( 19^{8} \) |
$13.35592$ |
$(a^3-2a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.092873899$ |
$401.1460858$ |
4.163315210 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( 1\) , \( a^{3} - 4 a - 3\) , \( -3 a^{2} - 7 a - 3\) , \( -a^{3} + 5 a^{2} + 21 a + 16\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){y}={x}^{3}+{x}^{2}+\left(-3a^{2}-7a-3\right){x}-a^{3}+5a^{2}+21a+16$ |
| 361.3-b1 |
361.3-b |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
361.3 |
\( 19^{2} \) |
\( 19^{8} \) |
$13.35592$ |
$(a^3-2a^2-3a+2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
|
\( 2 \) |
$1$ |
$15.02821138$ |
3.368042033 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( 6 a^{3} - 20 a^{2} - 6 a + 46\) , \( 14 a^{3} - 51 a^{2} - 6 a + 108\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(6a^{3}-20a^{2}-6a+46\right){x}+14a^{3}-51a^{2}-6a+108$ |
| 475.1-a1 |
475.1-a |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.1 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (a^3-2a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.125109672$ |
$316.7797274$ |
4.428853540 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( -a^{3} + 6 a + 5\) , \( a^{2} - 4\) , \( -11 a^{2} + 17 a + 35\) , \( 4 a^{3} - 26 a^{2} + 19 a + 57\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+6a+5\right){x}^{2}+\left(-11a^{2}+17a+35\right){x}+4a^{3}-26a^{2}+19a+57$ |
| 475.1-d1 |
475.1-d |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.1 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (a^3-2a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$72.31627871$ |
2.020315197 |
\( -\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361} \) |
\( \bigl[0\) , \( -a^{2} + 3\) , \( a^{2} - a - 4\) , \( -3 a^{3} - 4 a^{2} + 9 a + 12\) , \( -6 a^{3} - 6 a^{2} + 19 a + 20\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-3a^{3}-4a^{2}+9a+12\right){x}-6a^{3}-6a^{2}+19a+20$ |
*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.
