| 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 |
| 4275.2-a1 |
4275.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{10} \cdot 5^{2} \cdot 19^{4} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$3$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.194525286$ |
$1.577532480$ |
5.632063733 |
\( \frac{756058031}{438615} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 19\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3+19{x}$ |
| 4275.2-a2 |
4275.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{20} \cdot 5^{4} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$3$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.097262643$ |
$0.788766240$ |
5.632063733 |
\( \frac{48587168449}{28048275} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -76\) , \( -19\bigr] \) |
${y}^2+{x}{y}={x}^3-76{x}-19$ |
| 4275.2-b1 |
4275.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{2} \cdot 5^{6} \cdot 19^{4} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.310537232$ |
$1.959065350$ |
4.153795252 |
\( \frac{357911}{135375} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 2\) , \( -17\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2+2{x}-17$ |
| 4275.2-b2 |
4275.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{4} \cdot 5^{12} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.577634308$ |
$0.979532675$ |
4.153795252 |
\( \frac{90458382169}{2671875} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -93\) , \( -378\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-93{x}-378$ |
| 4275.2-c1 |
4275.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{16} \cdot 5^{6} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.885163107$ |
3.655266190 |
\( \frac{1256216039}{15582375} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 23\) , \( -176\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2+23{x}-176$ |
| 4275.2-c2 |
4275.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{4} \cdot 5^{24} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \cdot 3^{2} \) |
$1$ |
$0.221290776$ |
3.655266190 |
\( \frac{209595169258201}{41748046875} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -1237\) , \( 13054\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-1237{x}+13054$ |
| 4275.2-c3 |
4275.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{8} \cdot 5^{12} \cdot 19^{4} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$1$ |
$0.442581553$ |
3.655266190 |
\( \frac{6189976379881}{456890625} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -382\) , \( -2849\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-382{x}-2849$ |
| 4275.2-c4 |
4275.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{4} \cdot 5^{6} \cdot 19^{8} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.221290776$ |
3.655266190 |
\( \frac{23977812996389881}{146611125} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6007\) , \( -181724\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-6007{x}-181724$ |
| 4275.2-d1 |
4275.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{4} \cdot 5^{12} \cdot 19 \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.379648660$ |
$1.476202685$ |
5.142935224 |
\( \frac{71667639881}{185546875} a - \frac{254365484399}{333984375} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -8 a + 4\) , \( a + 66\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+a{x}^2+\left(-8a+4\right){x}+a+66$ |
| 4275.2-d2 |
4275.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{2} \cdot 5^{6} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 5 \) |
$0.189824330$ |
$2.952405371$ |
5.142935224 |
\( -\frac{381678169}{178125} a + \frac{9732964}{35625} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 2 a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+a{x}^2+\left(2a-1\right){x}$ |
| 4275.2-e1 |
4275.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{2} \cdot 5^{6} \cdot 19^{2} \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 5 \) |
$0.189824330$ |
$2.952405371$ |
5.142935224 |
\( \frac{381678169}{178125} a - \frac{333013349}{178125} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -4 a + 5\) , \( 5 a + 6\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(-4a+5\right){x}+5a+6$ |
| 4275.2-e2 |
4275.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4275.2 |
\( 3^{2} \cdot 5^{2} \cdot 19 \) |
\( 3^{4} \cdot 5^{12} \cdot 19 \) |
$3.14956$ |
$(-a), (a-1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.379648660$ |
$1.476202685$ |
5.142935224 |
\( -\frac{71667639881}{185546875} a - \frac{626818663066}{1669921875} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 6 a\) , \( -a + 28\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+6a{x}-a+28$ |
*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.
