| 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 |
| 25.2-a1 |
25.2-a |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$172.5411210$ |
1.349089582 |
\( \frac{37687449}{125} a^{3} - \frac{19419931}{125} a^{2} - \frac{43245961}{25} a + \frac{73603996}{125} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{3} - 6 a - 1\) , \( -123 a^{3} - 42 a^{2} + 724 a + 375\) , \( -109 a^{3} - 39 a^{2} + 644 a + 332\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-123a^{3}-42a^{2}+724a+375\right){x}-109a^{3}-39a^{2}+644a+332$ |
| 25.2-a2 |
25.2-a |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$172.5411210$ |
1.349089582 |
\( \frac{50670520594}{1953125} a^{3} - \frac{100307515486}{1953125} a^{2} - \frac{13618694406}{390625} a + \frac{8341092601}{1953125} \) |
\( \bigl[a^{3} - 6 a\) , \( a^{3} - 7 a\) , \( a^{2} + a - 2\) , \( 5 a^{3} + 2 a^{2} - 24 a + 1\) , \( a^{2} + 4 a + 6\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-7a\right){x}^{2}+\left(5a^{3}+2a^{2}-24a+1\right){x}+a^{2}+4a+6$ |
| 25.2-b1 |
25.2-b |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.086899803$ |
$229.6464956$ |
2.496587376 |
\( -\frac{716724786161}{390625} a^{3} + \frac{1442006043167}{390625} a^{2} + \frac{880238515792}{390625} a - \frac{24734039638}{390625} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( -6 a^{2} + 14 a - 1\) , \( -15 a^{3} + 33 a^{2} + 13 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-6a^{2}+14a-1\right){x}-15a^{3}+33a^{2}+13a-7$ |
| 25.2-b2 |
25.2-b |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.043449901$ |
$918.5859824$ |
2.496587376 |
\( \frac{194297}{625} a^{3} - \frac{281609}{625} a^{2} + \frac{263541}{625} a + \frac{1172301}{625} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( -a^{2} - a + 4\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-a^{2}-a+4\right){x}-2a+2$ |
| 25.2-c1 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{16} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$19.76098557$ |
2.472161035 |
\( \frac{9389615014508350332}{390625} a^{3} - \frac{4823466677992809061}{390625} a^{2} - \frac{53859864331393680042}{390625} a + \frac{18278320539983543932}{390625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -102 a^{3} - 21 a^{2} + 795 a - 268\) , \( -1965 a^{3} + 1912 a^{2} + 8713 a - 3073\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-102a^{3}-21a^{2}+795a-268\right){x}-1965a^{3}+1912a^{2}+8713a-3073$ |
| 25.2-c2 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1264.703077$ |
2.472161035 |
\( -\frac{632164}{25} a^{3} - \frac{867543}{5} a^{2} + \frac{12706826}{25} a + \frac{7522477}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 3 a^{3} - 6 a^{2} - 5 a - 3\) , \( -9 a^{3} + 19 a^{2} + 7 a - 5\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(3a^{3}-6a^{2}-5a-3\right){x}-9a^{3}+19a^{2}+7a-5$ |
| 25.2-c3 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$316.1757692$ |
2.472161035 |
\( -\frac{12122311834764}{125} a^{3} + \frac{140632327637578}{625} a^{2} + \frac{37375658851457}{625} a - \frac{26121479320169}{625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 33 a^{3} - 91 a^{2} + 20 a - 3\) , \( -524 a^{3} + 1171 a^{2} + 446 a - 263\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(33a^{3}-91a^{2}+20a-3\right){x}-524a^{3}+1171a^{2}+446a-263$ |
| 25.2-c4 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$79.04394231$ |
2.472161035 |
\( -\frac{5435814121359579324398236}{25} a^{3} + \frac{2522413903749638819346713}{5} a^{2} + \frac{3352607911018460603904474}{25} a - \frac{2342841126751495875736652}{25} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 3\) , \( a^{3} - 6 a\) , \( -128 a^{3} + 15 a^{2} + 856 a - 217\) , \( -1453 a^{3} + 1120 a^{2} + 7237 a - 2399\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-128a^{3}+15a^{2}+856a-217\right){x}-1453a^{3}+1120a^{2}+7237a-2399$ |
| 25.2-d1 |
25.2-d |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{2} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 1 \) |
$1$ |
$105.4074807$ |
6.378573759 |
\( \frac{1735607832}{5} a^{3} + \frac{582935254}{5} a^{2} - \frac{10217847707}{5} a - \frac{5167465863}{5} \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{2} - a + 2\) , \( a^{2} - 2\) , \( -5 a^{3} + a^{2} + 33 a - 11\) , \( 4 a^{3} - 6 a^{2} - 15 a + 5\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-5a^{3}+a^{2}+33a-11\right){x}+4a^{3}-6a^{2}-15a+5$ |
| 25.2-e1 |
25.2-e |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{2} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$229.5262754$ |
1.794653385 |
\( \frac{1735607832}{5} a^{3} + \frac{582935254}{5} a^{2} - \frac{10217847707}{5} a - \frac{5167465863}{5} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -28 a^{3} + 14 a^{2} + 158 a - 54\) , \( -97 a^{3} + 49 a^{2} + 554 a - 189\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-28a^{3}+14a^{2}+158a-54\right){x}-97a^{3}+49a^{2}+554a-189$ |
| 25.2-f1 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{16} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.606791681$ |
$44.94626835$ |
3.411940230 |
\( \frac{9389615014508350332}{390625} a^{3} - \frac{4823466677992809061}{390625} a^{2} - \frac{53859864331393680042}{390625} a + \frac{18278320539983543932}{390625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -536 a^{3} + 263 a^{2} + 3111 a - 1068\) , \( 9807 a^{3} - 5106 a^{2} - 56036 a + 18963\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-536a^{3}+263a^{2}+3111a-1068\right){x}+9807a^{3}-5106a^{2}-56036a+18963$ |
| 25.2-f2 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.213583362$ |
$179.7850734$ |
3.411940230 |
\( -\frac{12122311834764}{125} a^{3} + \frac{140632327637578}{625} a^{2} + \frac{37375658851457}{625} a - \frac{26121479320169}{625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -26 a^{3} - 2 a^{2} + 186 a - 58\) , \( 155 a^{3} - 159 a^{2} - 669 a + 239\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-26a^{3}-2a^{2}+186a-58\right){x}+155a^{3}-159a^{2}-669a+239$ |
| 25.2-f3 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$2.427166725$ |
$11.23656708$ |
3.411940230 |
\( -\frac{5435814121359579324398236}{25} a^{3} + \frac{2522413903749638819346713}{5} a^{2} + \frac{3352607911018460603904474}{25} a - \frac{2342841126751495875736652}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( 84 a^{3} - 267 a^{2} + 141 a - 8\) , \( 2643 a^{3} - 5996 a^{2} - 2030 a + 1279\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(84a^{3}-267a^{2}+141a-8\right){x}+2643a^{3}-5996a^{2}-2030a+1279$ |
| 25.2-f4 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.606791681$ |
$719.1402936$ |
3.411940230 |
\( -\frac{632164}{25} a^{3} - \frac{867543}{5} a^{2} + \frac{12706826}{25} a + \frac{7522477}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -a^{3} - 2 a^{2} + 6 a + 2\) , \( -3 a^{2} + 3 a + 3\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-a^{3}-2a^{2}+6a+2\right){x}-3a^{2}+3a+3$ |
| 25.2-g1 |
25.2-g |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$74.91961994$ |
2.343169631 |
\( -\frac{716724786161}{390625} a^{3} + \frac{1442006043167}{390625} a^{2} + \frac{880238515792}{390625} a - \frac{24734039638}{390625} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 4\) , \( a^{3} - 5 a\) , \( 1\) , \( -9 a^{3} + 5 a^{2} + 61 a - 11\) , \( -14 a^{3} + 11 a^{2} + 91 a - 38\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-4\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-9a^{3}+5a^{2}+61a-11\right){x}-14a^{3}+11a^{2}+91a-38$ |
| 25.2-g2 |
25.2-g |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$299.6784797$ |
2.343169631 |
\( \frac{194297}{625} a^{3} - \frac{281609}{625} a^{2} + \frac{263541}{625} a + \frac{1172301}{625} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 4\) , \( a^{3} - 5 a\) , \( 1\) , \( a^{3} + a + 14\) , \( a^{3} + 2 a^{2} + 2 a + 6\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-4\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(a^{3}+a+14\right){x}+a^{3}+2a^{2}+2a+6$ |
| 25.2-h1 |
25.2-h |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.057121782$ |
$542.2560016$ |
2.906267159 |
\( \frac{50670520594}{1953125} a^{3} - \frac{100307515486}{1953125} a^{2} - \frac{13618694406}{390625} a + \frac{8341092601}{1953125} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 6 a - 3\) , \( 13 a^{3} + 9 a^{2} - 64 a - 34\) , \( -35 a^{3} - 2 a^{2} + 234 a + 113\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-6a-3\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(13a^{3}+9a^{2}-64a-34\right){x}-35a^{3}-2a^{2}+234a+113$ |
| 25.2-h2 |
25.2-h |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.171365346$ |
$542.2560016$ |
2.906267159 |
\( \frac{37687449}{125} a^{3} - \frac{19419931}{125} a^{2} - \frac{43245961}{25} a + \frac{73603996}{125} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 6 a + 1\) , \( a^{3} - 5 a - 1\) , \( -16 a^{3} - 3 a^{2} + 89 a + 44\) , \( -28 a^{3} + 2 a^{2} + 142 a + 68\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+6a+1\right){x}^{2}+\left(-16a^{3}-3a^{2}+89a+44\right){x}-28a^{3}+2a^{2}+142a+68$ |
*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.
