| 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 |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{2} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1448.056347$ |
3.61111309 |
\( -\frac{32217243917}{3} a^{4} + 18007168092 a^{3} + \frac{124525089365}{3} a^{2} - \frac{213146129860}{3} a + \frac{47603120438}{3} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 4 a + 2\) , \( -a^{4} + a^{3} + 5 a^{2} - 5 a - 3\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 2\) , \( -a^{4} + 5 a^{2} + a\) , \( 4 a^{4} + a^{3} - 19 a^{2} - 8 a + 3\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+4a+2\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+4a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-5a-3\right){x}^{2}+\left(-a^{4}+5a^{2}+a\right){x}+4a^{4}+a^{3}-19a^{2}-8a+3$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{2} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.018468321$ |
$10888.10546$ |
2.50729464 |
\( \frac{747793342}{3} a^{4} - 753884124 a^{3} + \frac{839874611}{3} a^{2} + \frac{1290423893}{3} a - \frac{369335632}{3} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 4 a + 3\) , \( -2 a^{4} + a^{3} + 9 a^{2} - 3 a - 2\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 2\) , \( -6 a^{4} + a^{3} + 26 a^{2} - 6 a - 11\) , \( -3 a^{4} - 4 a^{3} + 7 a^{2} + 6 a + 1\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+4a+3\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+4a+2\right){y}={x}^{3}+\left(-2a^{4}+a^{3}+9a^{2}-3a-2\right){x}^{2}+\left(-6a^{4}+a^{3}+26a^{2}-6a-11\right){x}-3a^{4}-4a^{3}+7a^{2}+6a+1$ |
| 9.1-c1 |
9.1-c |
$4$ |
$15$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{10} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 1 \) |
$5.024095054$ |
$1.191540139$ |
1.86608819 |
\( -\frac{1362859584417210265552}{243} a^{4} - \frac{182839759416657436177}{27} a^{3} + \frac{3182056032798762423940}{243} a^{2} + \frac{1572554890893820547857}{243} a - \frac{617417963367380388536}{243} \) |
\( \bigl[a^{4} - 5 a^{2} + 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 2 a^{4} - a^{3} - 9 a^{2} + 5 a + 2\) , \( 546 a^{4} - 878 a^{3} - 2142 a^{2} + 3424 a - 759\) , \( 13177 a^{4} - 21774 a^{3} - 51256 a^{2} + 85685 a - 18507\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+2\right){x}{y}+\left(2a^{4}-a^{3}-9a^{2}+5a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(546a^{4}-878a^{3}-2142a^{2}+3424a-759\right){x}+13177a^{4}-21774a^{3}-51256a^{2}+85685a-18507$ |
| 9.1-c2 |
9.1-c |
$4$ |
$15$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{2} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1.004819010$ |
$3723.562937$ |
1.86608819 |
\( -\frac{643713008378}{3} a^{4} + 156600161332 a^{3} + \frac{3345490169651}{3} a^{2} - \frac{1670996131882}{3} a - \frac{2382588076453}{3} \) |
\( \bigl[a^{4} - 4 a^{2} + a + 1\) , \( -a^{4} + 5 a^{2} - a - 3\) , \( a^{4} - 5 a^{2} + 2\) , \( 7 a^{4} - 9 a^{3} - 26 a^{2} + 38 a - 7\) , \( -13 a^{4} + 24 a^{3} + 51 a^{2} - 93 a + 20\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a+1\right){x}{y}+\left(a^{4}-5a^{2}+2\right){y}={x}^{3}+\left(-a^{4}+5a^{2}-a-3\right){x}^{2}+\left(7a^{4}-9a^{3}-26a^{2}+38a-7\right){x}-13a^{4}+24a^{3}+51a^{2}-93a+20$ |
| 9.1-c3 |
9.1-c |
$4$ |
$15$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{6} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$0.334939670$ |
$11170.68881$ |
1.86608819 |
\( \frac{12379375}{27} a^{4} - \frac{37575164}{27} a^{3} + \frac{14317900}{27} a^{2} + \frac{21228419}{27} a - \frac{6342761}{27} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 4 a\) , \( -a^{4} + 6 a^{2} - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 5 a\) , \( -13 a^{4} + 8 a^{3} + 70 a^{2} - 28 a - 51\) , \( 54 a^{4} - 40 a^{3} - 279 a^{2} + 143 a + 197\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+4a\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+5a\right){y}={x}^{3}+\left(-a^{4}+6a^{2}-3\right){x}^{2}+\left(-13a^{4}+8a^{3}+70a^{2}-28a-51\right){x}+54a^{4}-40a^{3}-279a^{2}+143a+197$ |
| 9.1-c4 |
9.1-c |
$4$ |
$15$ |
5.5.160801.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{30} \) |
$44.63825$ |
$(a^4-5a^2+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 1 \) |
$1.674698351$ |
$3.574620419$ |
1.86608819 |
\( -\frac{268605305861390829836407}{14348907} a^{4} + \frac{450394293457823967148907}{14348907} a^{3} + \frac{1038204841696182050112038}{14348907} a^{2} - \frac{1777066342417430001622352}{14348907} a + \frac{396882183492690134473523}{14348907} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 4 a\) , \( -a^{4} + 6 a^{2} - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 5 a\) , \( -18 a^{4} - 42 a^{3} + 20 a^{2} + 67 a + 9\) , \( -3353 a^{4} + 1876 a^{3} + 16593 a^{2} - 7586 a - 11486\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+4a\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+5a\right){y}={x}^{3}+\left(-a^{4}+6a^{2}-3\right){x}^{2}+\left(-18a^{4}-42a^{3}+20a^{2}+67a+9\right){x}-3353a^{4}+1876a^{3}+16593a^{2}-7586a-11486$ |
*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.
