| 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 |
| 400.1-a1 |
400.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{12} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.971012428$ |
$5.171827352$ |
2.622606252 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 13371 a - 45087\) , \( -2153934 a + 7263670\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(13371a-45087\right){x}-2153934a+7263670$ |
| 400.1-a2 |
400.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{4} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.913037286$ |
$5.171827352$ |
2.622606252 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 1349 a + 3204\) , \( -45306 a - 107480\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(1349a+3204\right){x}-45306a-107480$ |
| 400.1-a3 |
400.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{2} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1.456518643$ |
$20.68730941$ |
2.622606252 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -491 a - 1161\) , \( -7281 a - 17274\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-491a-1161\right){x}-7281a-17274$ |
| 400.1-a4 |
400.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{6} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3^{3} \) |
$0.485506214$ |
$20.68730941$ |
2.622606252 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 15211 a - 51292\) , \( -1718199 a + 5794249\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(15211a-51292\right){x}-1718199a+5794249$ |
| 400.1-b1 |
400.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{8} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.602978083$ |
$5.511161689$ |
3.470874898 |
\( -\frac{3538944}{625} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -64 a - 152\) , \( 540 a + 1281\bigr] \) |
${y}^2={x}^{3}+\left(-64a-152\right){x}+540a+1281$ |
| 400.1-b2 |
400.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{4} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.205956167$ |
$5.511161689$ |
3.470874898 |
\( \frac{1016339184}{25} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1064 a - 2527\) , \( 32040 a + 76006\bigr] \) |
${y}^2={x}^{3}+\left(-1064a-2527\right){x}+32040a+76006$ |
| 400.1-c1 |
400.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{12} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$4.977298798$ |
$0.886343882$ |
2.303882094 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -36\) , \( -140\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-36{x}-140$ |
| 400.1-c2 |
400.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{4} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1.659099599$ |
$7.977094942$ |
2.303882094 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 4\) , \( 4\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+4{x}+4$ |
| 400.1-c3 |
400.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{2} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.829549799$ |
$31.90837977$ |
2.303882094 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-{x}$ |
| 400.1-c4 |
400.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{6} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$2.488649399$ |
$3.545375530$ |
2.303882094 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -41\) , \( -116\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-41{x}-116$ |
| 400.1-d1 |
400.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 5^{8} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.602978083$ |
$5.511161689$ |
3.470874898 |
\( -\frac{3538944}{625} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 64 a - 216\) , \( -540 a + 1821\bigr] \) |
${y}^2={x}^{3}+\left(64a-216\right){x}-540a+1821$ |
| 400.1-d2 |
400.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
400.1 |
\( 2^{4} \cdot 5^{2} \) |
\( 2^{16} \cdot 5^{4} \) |
$2.29568$ |
$(-a-2), (-a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.205956167$ |
$5.511161689$ |
3.470874898 |
\( \frac{1016339184}{25} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1064 a - 3591\) , \( -32040 a + 108046\bigr] \) |
${y}^2={x}^{3}+\left(1064a-3591\right){x}-32040a+108046$ |
*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.
