| 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 |
| 4.1-a2 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.868769197$ |
0.336733136 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36 a - 86\) , \( -2492 a - 5912\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-36a-86\right){x}-2492a-5912$ |
| 4.1-b2 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/18\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$13.40768290$ |
1.166989006 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 44 a - 145\) , \( -242 a + 809\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(44a-145\right){x}-242a+809$ |
| 128.5-c2 |
128.5-c |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( - 2^{45} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.546351081$ |
3.989365447 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 247 a - 825\) , \( 3371 a - 11359\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(247a-825\right){x}+3371a-11359$ |
| 128.5-g2 |
128.5-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( - 2^{45} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.009192170$ |
$2.546351081$ |
2.667726059 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -409 a - 969\) , \( -95979 a - 227689\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-409a-969\right){x}-95979a-227689$ |
| 128.6-c2 |
128.6-c |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( - 2^{45} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.546351081$ |
3.989365447 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 496 a - 1678\) , \( -10230 a + 34493\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(496a-1678\right){x}-10230a+34493$ |
| 128.6-g2 |
128.6-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( - 2^{45} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.504596085$ |
$2.546351081$ |
2.667726059 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -203 a - 478\) , \( 33067 a + 78444\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-203a-478\right){x}+33067a+78444$ |
| 256.1-e2 |
256.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{51} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.967192299$ |
3.030598229 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 699 a - 2355\) , \( 16370 a - 55214\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(699a-2355\right){x}+16370a-55214$ |
| 256.1-j2 |
256.1-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{51} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.351920727$ |
2.333978012 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -576 a - 1368\) , \( 159488 a + 378352\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-576a-1368\right){x}+159488a+378352$ |
| 288.3-g2 |
288.3-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{39} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.870464668$ |
1.347522833 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 115 a - 392\) , \( -1115 a + 3816\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(115a-392\right){x}-1115a+3816$ |
| 288.3-j2 |
288.3-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{39} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.676588352$ |
$1.116817468$ |
4.735351769 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -1958 a - 4648\) , \( -998515 a - 2368760\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-1958a-4648\right){x}-998515a-2368760$ |
| 288.4-g2 |
288.4-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{39} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.870464668$ |
1.347522833 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 2374 a - 8006\) , \( -104500 a + 352404\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2374a-8006\right){x}-104500a+352404$ |
| 288.4-j2 |
288.4-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{39} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$0.338294176$ |
$1.116817468$ |
4.735351769 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -96 a - 234\) , \( -10746 a - 25503\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-96a-234\right){x}-10746a-25503$ |
| 484.1-a2 |
484.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
484.1 |
\( 2^{2} \cdot 11^{2} \) |
\( - 2^{27} \cdot 11^{6} \) |
$2.40772$ |
$(-a-2), (-a+3), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.171535498$ |
3.402142284 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 22093 a - 74505\) , \( -2998592 a + 10112093\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(22093a-74505\right){x}-2998592a+10112093$ |
| 484.1-j2 |
484.1-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
484.1 |
\( 2^{2} \cdot 11^{2} \) |
\( - 2^{27} \cdot 11^{6} \) |
$2.40772$ |
$(-a-2), (-a+3), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.171535498$ |
3.402142284 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 2 a - 53\) , \( 340 a + 577\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(2a-53\right){x}+340a+577$ |
*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.
