| 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 |
| 175.2-a1 |
175.2-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( - 5^{6} \cdot 7^{9} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$1$ |
$4.761309222$ |
1.582766718 |
\( -\frac{16146410084579303}{40353607} a^{2} - \frac{24513786018922996}{40353607} a + \frac{51294295430574624}{40353607} \) |
\( \bigl[a\) , \( a^{2} + a - 5\) , \( a^{2} + a - 5\) , \( 33 a^{2} + a - 225\) , \( -218 a^{2} - 51 a + 1450\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(33a^{2}+a-225\right){x}-218a^{2}-51a+1450$ |
| 175.2-a2 |
175.2-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( - 5^{6} \cdot 7^{3} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$14.28392766$ |
1.582766718 |
\( -\frac{470436}{343} a^{2} + \frac{1143587}{343} a - \frac{168936}{343} \) |
\( \bigl[a\) , \( a^{2} + a - 5\) , \( a^{2} + a - 5\) , \( -2 a^{2} + a + 15\) , \( a^{2} + a - 7\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(-2a^{2}+a+15\right){x}+a^{2}+a-7$ |
| 175.2-b1 |
175.2-b |
$4$ |
$4$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{7} \cdot 7 \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.067183206$ |
$18.82602717$ |
4.312279647 |
\( -\frac{1750534952787}{35} a^{2} - \frac{311934841324}{35} a + \frac{11886225135907}{35} \) |
\( \bigl[a^{2} - 5\) , \( -1\) , \( 0\) , \( -951439 a^{2} - 1444473 a + 3022603\) , \( -902873569 a^{2} - 1370741478 a + 2868315576\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}={x}^{3}-{x}^{2}+\left(-951439a^{2}-1444473a+3022603\right){x}-902873569a^{2}-1370741478a+2868315576$ |
| 175.2-b2 |
175.2-b |
$4$ |
$4$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{7} \cdot 7 \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.516795801$ |
$75.30410869$ |
4.312279647 |
\( \frac{43928057}{35} a^{2} + \frac{66690044}{35} a - \frac{139559367}{35} \) |
\( \bigl[1\) , \( a^{2} - 5\) , \( a^{2} + a - 4\) , \( 721658 a^{2} + 129994 a - 4903620\) , \( 1648726004 a^{2} + 293497360 a - 11194191591\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(721658a^{2}+129994a-4903620\right){x}+1648726004a^{2}+293497360a-11194191591$ |
| 175.2-b3 |
175.2-b |
$4$ |
$4$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( - 5^{10} \cdot 7^{4} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.516795801$ |
$37.65205434$ |
4.312279647 |
\( \frac{2809194534917279}{1500625} a^{2} - \frac{10383887586997322}{1500625} a + \frac{8334678625039421}{1500625} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -1215 a^{2} + 4496 a - 3616\) , \( 74796 a^{2} - 276474 a + 221911\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1215a^{2}+4496a-3616\right){x}+74796a^{2}-276474a+221911$ |
| 175.2-b4 |
175.2-b |
$4$ |
$4$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{8} \cdot 7^{2} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.033591603$ |
$75.30410869$ |
4.312279647 |
\( -\frac{107909754}{1225} a^{2} - \frac{64204353}{1225} a + \frac{850593529}{1225} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -75 a^{2} + 281 a - 231\) , \( 1232 a^{2} - 4551 a + 3648\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-75a^{2}+281a-231\right){x}+1232a^{2}-4551a+3648$ |
| 175.2-c1 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{6} \cdot 7^{8} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.074458170$ |
$36.20090687$ |
3.584118192 |
\( \frac{1824928468}{5764801} a^{2} - \frac{1345230953}{5764801} a - \frac{2105854007}{5764801} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} + a - 4\) , \( 0\) , \( 48039478 a^{2} + 72933473 a - 152615368\) , \( 270822288210 a^{2} + 411162045709 a - 860368288753\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(48039478a^{2}+72933473a-152615368\right){x}+270822288210a^{2}+411162045709a-860368288753$ |
| 175.2-c2 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( - 5^{6} \cdot 7^{2} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.074458170$ |
$144.8036275$ |
3.584118192 |
\( \frac{24849000}{49} a^{2} + \frac{37748161}{49} a - \frac{78881609}{49} \) |
\( \bigl[1\) , \( -a^{2} - a + 4\) , \( a\) , \( 215627501 a^{2} + 38423545 a - 1464122111\) , \( -451635498322 a^{2} - 80478772129 a + 3066628875362\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(215627501a^{2}+38423545a-1464122111\right){x}-451635498322a^{2}-80478772129a+3066628875362$ |
| 175.2-c3 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{6} \cdot 7^{4} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.074458170$ |
$144.8036275$ |
3.584118192 |
\( -\frac{225573007}{2401} a^{2} - \frac{58067834}{2401} a + \frac{1583231073}{2401} \) |
\( \bigl[1\) , \( -a^{2} + a + 6\) , \( a\) , \( 115 a^{2} + 19 a - 779\) , \( -877 a^{2} - 156 a + 5952\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(115a^{2}+19a-779\right){x}-877a^{2}-156a+5952$ |
| 175.2-c4 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{6} \cdot 7 \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.191330722$ |
$18.10045343$ |
3.584118192 |
\( -\frac{84096891433574753384471}{7} a^{2} - \frac{14985568201611275023622}{7} a + \frac{571022332294137577763767}{7} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 4\) , \( a^{2} + a - 4\) , \( 24 a^{2} + 52 a - 250\) , \( -245 a^{2} - 473 a + 1667\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(24a^{2}+52a-250\right){x}-245a^{2}-473a+1667$ |
| 175.2-c5 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{6} \cdot 7^{2} \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.297832680$ |
$72.40181375$ |
3.584118192 |
\( -\frac{2353024794450}{49} a^{2} - \frac{434940018425}{49} a + \frac{16016561377497}{49} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 4\) , \( a^{2} + a - 4\) , \( -21 a^{2} + 22 a - 5\) , \( -30 a^{2} - 338 a + 452\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-21a^{2}+22a-5\right){x}-30a^{2}-338a+452$ |
| 175.2-c6 |
175.2-c |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( - 5^{6} \cdot 7 \) |
$5.72177$ |
$(-a+3), (a^2+2a-3)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.191330722$ |
$18.10045343$ |
3.584118192 |
\( \frac{960431553635305495}{7} a^{2} - \frac{3550132253411453690}{7} a + \frac{2849530108248409337}{7} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 4\) , \( a^{2} + a - 4\) , \( -386 a^{2} + 312 a + 160\) , \( 29 a^{2} - 20035 a + 23577\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-386a^{2}+312a+160\right){x}+29a^{2}-20035a+23577$ |
*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.
