| 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 |
| 16.1-a1 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$3.848457779$ |
$1.923527708$ |
2.397348384 |
\( -\frac{573746144654542909099}{2} a^{3} + 448591720203712073410 a^{2} + 1181481234689977876243 a - \frac{3053308720069032929009}{2} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 4 a - 1\) , \( -27 a^{3} + 38 a^{2} + 129 a - 152\) , \( -225 a^{3} + 384 a^{2} + 950 a - 1419\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(-27a^{3}+38a^{2}+129a-152\right){x}-225a^{3}+384a^{2}+950a-1419$ |
| 16.1-a2 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1.282819259$ |
$155.8057444$ |
2.397348384 |
\( -\frac{3615673}{2} a^{3} + \frac{22615701}{8} a^{2} + \frac{14891109}{2} a - 9620794 \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 4 a - 1\) , \( 3 a^{3} - 2 a^{2} - 11 a + 8\) , \( a^{3} - 2 a + 3\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(3a^{3}-2a^{2}-11a+8\right){x}+a^{3}-2a+3$ |
| 16.1-a3 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.427606419$ |
$1402.251699$ |
2.397348384 |
\( \frac{6596973}{2} a^{3} - \frac{3508603}{2} a^{2} - \frac{43232855}{2} a - \frac{22074779}{2} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{2} + 4\) , \( 0\) , \( a^{3} - 2 a + 5\) , \( -5 a^{3} + 9 a^{2} + 21 a - 27\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(a^{3}-2a+5\right){x}-5a^{3}+9a^{2}+21a-27$ |
| 16.1-b1 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
|
\( 1 \) |
$1$ |
$475.9227258$ |
1.518617981 |
\( -\frac{573746144654542909099}{2} a^{3} + 448591720203712073410 a^{2} + 1181481234689977876243 a - \frac{3053308720069032929009}{2} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 4 a - 2\) , \( a^{2} + a - 3\) , \( -81 a^{3} + 102 a^{2} + 343 a - 318\) , \( -257 a^{3} - 912 a^{2} + 1591 a + 4663\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(-81a^{3}+102a^{2}+343a-318\right){x}-257a^{3}-912a^{2}+1591a+4663$ |
| 16.1-b2 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
|
\( 1 \) |
$1$ |
$52.88030287$ |
1.518617981 |
\( \frac{6596973}{2} a^{3} - \frac{3508603}{2} a^{2} - \frac{43232855}{2} a - \frac{22074779}{2} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - 2\) , \( 0\) , \( 2 a^{3} + 4 a^{2} - 6 a - 4\) , \( 4 a^{3} - 2 a^{2} - 4 a\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(2a^{3}+4a^{2}-6a-4\right){x}+4a^{3}-2a^{2}-4a$ |
| 16.1-b3 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
|
\( 3 \) |
$1$ |
$475.9227258$ |
1.518617981 |
\( -\frac{3615673}{2} a^{3} + \frac{22615701}{8} a^{2} + \frac{14891109}{2} a - 9620794 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( -a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-a^{3}+a^{2}+3a-3\right){x}-a^{2}+1$ |
| 16.1-c1 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.628139834$ |
$729.6575744$ |
4.749760984 |
\( \frac{54906875467771}{2} a^{3} + 77871175883534 a^{2} + 52479510197237 a + \frac{19511461081835}{2} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 2 a^{2} + 5 a - 6\) , \( a^{3} - 3 a - 1\) , \( -5 a^{3} - a^{2} + 15 a - 8\) , \( 2380 a^{3} - 3719 a^{2} - 9802 a + 12660\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-6\right){x}^{2}+\left(-5a^{3}-a^{2}+15a-8\right){x}+2380a^{3}-3719a^{2}-9802a+12660$ |
| 16.1-c2 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.628139834$ |
$81.07306383$ |
4.749760984 |
\( -1495210 a^{3} + \frac{4635579}{2} a^{2} + \frac{12430071}{2} a - 7948001 \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 3 a\) , \( -2 a^{3} - 3 a^{2} + 3 a + 3\) , \( -4 a^{3} - 7 a^{2} + 3 a + 2\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-2a^{3}-3a^{2}+3a+3\right){x}-4a^{3}-7a^{2}+3a+2$ |
| 16.1-c3 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.542713278$ |
$729.6575744$ |
4.749760984 |
\( -\frac{6035}{2} a^{3} + \frac{59547}{8} a^{2} + \frac{135879}{4} a + \frac{64975}{4} \) |
\( \bigl[a\) , \( 1\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 10 a - 4\) , \( -8 a^{3} + 9 a^{2} + 34 a - 27\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+{x}^{2}+\left(2a^{3}-10a-4\right){x}-8a^{3}+9a^{2}+34a-27$ |
| 16.1-d1 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.081497900$ |
$6.194235184$ |
2.169498472 |
\( \frac{54906875467771}{2} a^{3} + 77871175883534 a^{2} + 52479510197237 a + \frac{19511461081835}{2} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a + 1\) , \( 0\) , \( 18 a^{3} - 81 a - 45\) , \( 73 a^{3} + 95 a^{2} - 370 a - 607\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(18a^{3}-81a-45\right){x}+73a^{3}+95a^{2}-370a-607$ |
| 16.1-d2 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.360499300$ |
$501.7330499$ |
2.169498472 |
\( -\frac{6035}{2} a^{3} + \frac{59547}{8} a^{2} + \frac{135879}{4} a + \frac{64975}{4} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a + 1\) , \( 0\) , \( -2 a^{3} + 9 a + 5\) , \( a^{3} - 3 a^{2} - 4 a + 13\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(-2a^{3}+9a+5\right){x}+a^{3}-3a^{2}-4a+13$ |
| 16.1-d3 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.120166433$ |
$4515.597449$ |
2.169498472 |
\( -1495210 a^{3} + \frac{4635579}{2} a^{2} + \frac{12430071}{2} a - 7948001 \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a + 1\) , \( 10 a^{3} - 29 a^{2} + 3 a + 21\) , \( -51 a^{3} + 149 a^{2} - 35 a - 81\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(10a^{3}-29a^{2}+3a+21\right){x}-51a^{3}+149a^{2}-35a-81$ |
*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.
