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 |
33.1-a1 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11 \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$5.915402008$ |
$4.468126413$ |
2.300502718 |
\( -\frac{181064514814}{99} a + \frac{610600481027}{99} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 943 a - 3179\) , \( 27552 a - 92919\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(943a-3179\right){x}+27552a-92919$ |
33.1-a2 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.957701004$ |
$2.234063206$ |
2.300502718 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
33.1-a3 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.478850502$ |
$8.936252827$ |
2.300502718 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
33.1-a4 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.957701004$ |
$8.936252827$ |
2.300502718 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
33.1-a5 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.739425251$ |
$8.936252827$ |
2.300502718 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
33.1-a6 |
33.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11 \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$5.915402008$ |
$4.468126413$ |
2.300502718 |
\( \frac{181064514814}{99} a + \frac{429535966213}{99} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -944 a - 2235\) , \( -27553 a - 65366\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-944a-2235\right){x}-27553a-65366$ |
33.1-b1 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11 \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$30.13204906$ |
0.327832279 |
\( -\frac{181064514814}{99} a + \frac{610600481027}{99} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -a - 1\) , \( 9 a + 24\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a-1\right){x}+9a+24$ |
33.1-b2 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.883253066$ |
0.327832279 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -16001 a + 53960\) , \( 639500 a - 2156578\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16001a+53960\right){x}+639500a-2156578$ |
33.1-b3 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$7.533012266$ |
0.327832279 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 4239 a - 14295\) , \( 75585 a - 254898\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4239a-14295\right){x}+75585a-254898$ |
33.1-b4 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$30.13204906$ |
0.327832279 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 2399 a - 8090\) , \( -110030 a + 371048\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2399a-8090\right){x}-110030a+371048$ |
33.1-b5 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.883253066$ |
0.327832279 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -53920 a - 127910\) , \( -11482951 a - 27240791\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-53920a-127910\right){x}-11482951a-27240791$ |
33.1-b6 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11 \) |
$1.23034$ |
$(-2a+7), (-4a-9)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$30.13204906$ |
0.327832279 |
\( \frac{181064514814}{99} a + \frac{429535966213}{99} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -1\) , \( -10 a + 34\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}-{x}-10a+34$ |
*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.