| 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 |
| 253.1-A1 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( - 11^{2} \cdot 23 \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$40.38798431$ |
0.526342305 |
\( \frac{417863}{2783} a^{2} - \frac{34767}{2783} a + \frac{3149985}{2783} \) |
\( \bigl[a^{2} + a + 1\) , \( -a^{2} - a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}+a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+{x}$ |
| 253.1-A2 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( 11^{4} \cdot 23^{2} \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$20.19399215$ |
0.526342305 |
\( \frac{2062387370025}{7745089} a^{2} - \frac{476450732000}{7745089} a - \frac{1516352690708}{7745089} \) |
\( \bigl[a + 1\) , \( a^{2} - a\) , \( 1\) , \( -5 a^{2} + 4 a + 5\) , \( -3 a^{2} - 3 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a\right){x}^{2}+\left(-5a^{2}+4a+5\right){x}-3a^{2}-3a-1$ |
| 253.1-A3 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( 11^{8} \cdot 23^{4} \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$5.048498038$ |
0.526342305 |
\( \frac{898671238793091074}{59986403617921} a^{2} - \frac{1600222393063032451}{59986403617921} a + \frac{1227014935973120994}{59986403617921} \) |
\( \bigl[a^{2} + a + 1\) , \( -a^{2} - a - 1\) , \( 0\) , \( -10 a^{2} + 15 a - 14\) , \( -23 a^{2} + 37 a - 18\bigr] \) |
${y}^2+\left(a^{2}+a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+\left(-10a^{2}+15a-14\right){x}-23a^{2}+37a-18$ |
| 253.1-A4 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( 11^{2} \cdot 23 \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$10.09699607$ |
0.526342305 |
\( -\frac{174588745152190}{2783} a^{2} - \frac{130042620481635}{2783} a + \frac{1327777229138}{2783} \) |
\( \bigl[a^{2} + a\) , \( -a\) , \( a^{2} + 1\) , \( 174 a^{2} - 85 a - 169\) , \( 1133 a^{2} - 241 a - 824\bigr] \) |
${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+1\right){y}={x}^{3}-a{x}^{2}+\left(174a^{2}-85a-169\right){x}+1133a^{2}-241a-824$ |
| 253.1-A5 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( 11^{4} \cdot 23^{8} \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.262124509$ |
0.526342305 |
\( \frac{833654495257217661469537}{1146551135499121} a^{2} - \frac{1463012109643929405934449}{1146551135499121} a + \frac{1104329696417202846546921}{1146551135499121} \) |
\( \bigl[a^{2} + 1\) , \( a - 1\) , \( 0\) , \( -449 a^{2} + 838 a - 639\) , \( -6934 a^{2} + 12248 a - 9210\bigr] \) |
${y}^2+\left(a^{2}+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-449a^{2}+838a-639\right){x}-6934a^{2}+12248a-9210$ |
| 253.1-A6 |
253.1-A |
$6$ |
$8$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
253.1 |
\( 11 \cdot 23 \) |
\( - 11^{16} \cdot 23^{2} \) |
$1.07776$ |
$(a^2+a-2), (-a^2-2a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.262124509$ |
0.526342305 |
\( \frac{12096844958864335269183}{24307407097829673169} a^{2} + \frac{81129911893400377438417}{24307407097829673169} a + \frac{54350390176978478717063}{24307407097829673169} \) |
\( \bigl[a + 1\) , \( a^{2} - a\) , \( 1\) , \( -41 a - 30\) , \( -82 a^{2} - 80 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a\right){x}^{2}+\left(-41a-30\right){x}-82a^{2}-80a-50$ |
*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.
