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 |
15.2-a1 |
15.2-a |
$4$ |
$10$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{10} \cdot 5^{10} \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.839847852$ |
$18.99910249$ |
1.708520240 |
\( \frac{805058590861631339}{576650390625} a^{2} + \frac{1268977584604502531}{576650390625} a - \frac{1562043547410087929}{576650390625} \) |
\( \bigl[a^{2} - 3\) , \( 1\) , \( a^{2} + a - 3\) , \( 75 a^{2} - 12 a - 472\) , \( 161 a^{2} - 42 a - 967\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+{x}^{2}+\left(75a^{2}-12a-472\right){x}+161a^{2}-42a-967$ |
15.2-a2 |
15.2-a |
$4$ |
$10$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5^{5} \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 1 \) |
$1.679695705$ |
$37.99820499$ |
1.708520240 |
\( -\frac{25089380164}{759375} a^{2} - \frac{129708562831}{759375} a + \frac{498426941254}{759375} \) |
\( \bigl[a^{2} - 3\) , \( 1\) , \( a^{2} + a - 3\) , \( 55 a^{2} - 7 a - 352\) , \( 438 a^{2} - 88 a - 2686\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+{x}^{2}+\left(55a^{2}-7a-352\right){x}+438a^{2}-88a-2686$ |
15.2-a3 |
15.2-a |
$4$ |
$10$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{2} \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.167969570$ |
$94.99551247$ |
1.708520240 |
\( -\frac{474614682832216}{225} a^{2} + \frac{88887613981061}{225} a + \frac{2919928520246476}{225} \) |
\( \bigl[a^{2} + a - 3\) , \( -1\) , \( a^{2} + a - 3\) , \( 20 a^{2} + 3 a - 135\) , \( -95 a^{2} + 40 a + 541\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}-{x}^{2}+\left(20a^{2}+3a-135\right){x}-95a^{2}+40a+541$ |
15.2-a4 |
15.2-a |
$4$ |
$10$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 1 \) |
$0.335939141$ |
$189.9910249$ |
1.708520240 |
\( -\frac{7431769}{15} a^{2} - \frac{4179976}{15} a + \frac{60073024}{15} \) |
\( \bigl[a^{2} + a - 3\) , \( -1\) , \( a^{2} + a - 3\) , \( 8 a - 15\) , \( -4 a^{2} + 17 a - 3\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}-{x}^{2}+\left(8a-15\right){x}-4a^{2}+17a-3$ |
15.2-b1 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.677329918$ |
0.957863551 |
\( -\frac{2762421153954670601047049}{3515625} a^{2} + \frac{517356524035998674755129}{3515625} a + \frac{16994990990732196688324364}{3515625} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( 255 a^{2} - 30 a - 1607\) , \( 4253 a^{2} - 756 a - 26252\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(255a^{2}-30a-1607\right){x}+4253a^{2}-756a-26252$ |
15.2-b2 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{8} \cdot 5^{2} \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$1.677329918$ |
0.957863551 |
\( \frac{2154621983559594094227929}{164025} a^{2} - \frac{7300914501119667671929609}{164025} a + \frac{4510427532245375321396356}{164025} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( -75 a^{2} + 300 a - 277\) , \( -1077 a^{2} + 3914 a - 2954\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-75a^{2}+300a-277\right){x}-1077a^{2}+3914a-2954$ |
15.2-b3 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$13.41863935$ |
0.957863551 |
\( \frac{27137864062334}{50625} a^{2} - \frac{152423185980739}{50625} a + \frac{212562263704501}{50625} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( 10 a^{2} + 15 a - 102\) , \( 74 a^{2} + 23 a - 549\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(10a^{2}+15a-102\right){x}+74a^{2}+23a-549$ |
15.2-b4 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$107.3491148$ |
0.957863551 |
\( \frac{140568755549}{225} a^{2} + \frac{221499293246}{225} a - \frac{272852338439}{225} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( 3\) , \( 2 a^{2} - 12\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+3{x}+2a^{2}-12$ |
15.2-b5 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$429.3964592$ |
0.957863551 |
\( -\frac{130201}{15} a^{2} - \frac{202429}{15} a + \frac{287176}{15} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( 8\) , \( 2 a + 1\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+8{x}+2a+1$ |
15.2-b6 |
15.2-b |
$6$ |
$8$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5 \) |
$3.93177$ |
$(a-2), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$26.83727870$ |
0.957863551 |
\( \frac{51241969707324211486}{15} a^{2} + \frac{80745736551235585069}{15} a - \frac{99469092005530821931}{15} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 3\) , \( 1\) , \( -10 a^{2} - 15 a + 28\) , \( 18 a^{2} + 29 a - 47\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-10a^{2}-15a+28\right){x}+18a^{2}+29a-47$ |
15.2-c1 |
15.2-c |
$2$ |
$2$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{10} \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.057798737$ |
$89.22615774$ |
2.761003606 |
\( \frac{1802836380554}{87890625} a^{2} - \frac{3740144876359}{87890625} a + \frac{2000168056831}{87890625} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a\) , \( -17 a^{2} + 55 a - 29\) , \( 108 a^{2} - 368 a + 230\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-17a^{2}+55a-29\right){x}+108a^{2}-368a+230$ |
15.2-c2 |
15.2-c |
$2$ |
$2$ |
3.3.785.1 |
$3$ |
$[3, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$3.93177$ |
$(a-2), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.115597474$ |
$178.4523154$ |
2.761003606 |
\( -\frac{1859341}{9375} a^{2} - \frac{2924314}{9375} a + \frac{19868551}{9375} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a\) , \( -2 a^{2} + 5 a + 1\) , \( -2 a + 3\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-2a^{2}+5a+1\right){x}-2a+3$ |
*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.