| 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 |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$2.146444646$ |
$2.234063206$ |
2.986504436 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
| 33.1-a2 |
33.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1.073222323$ |
$8.936252827$ |
2.986504436 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
| 33.1-a3 |
33.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.146444646$ |
$8.936252827$ |
2.986504436 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
| 33.1-a4 |
33.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$2.146444646$ |
$8.936252827$ |
2.986504436 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
| 33.1-b1 |
33.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{36} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1.536762106$ |
$1.883253066$ |
5.407351434 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 391\) , \( -1092\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+391{x}-1092$ |
| 33.1-b2 |
33.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{4} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$3.073524213$ |
$7.533012266$ |
5.407351434 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -104\) , \( -102\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-104{x}-102$ |
| 33.1-b3 |
33.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{18} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.536762106$ |
$30.13204906$ |
5.407351434 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -59\) , \( 186\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-59{x}+186$ |
| 33.1-b4 |
33.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{18} \cdot 11^{8} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$6.147048426$ |
$1.883253066$ |
5.407351434 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1319\) , \( -18084\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-1319{x}-18084$ |
| 33.1-c1 |
33.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.883253066$ |
0.586444209 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -557 a + 3963\) , \( 5853 a - 40304\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-557a+3963\right){x}+5853a-40304$ |
| 33.1-c2 |
33.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.533012266$ |
0.586444209 |
\( \frac{169112377}{88209} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 158 a - 987\) , \( 903 a - 6039\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(158a-987\right){x}+903a-6039$ |
| 33.1-c3 |
33.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$30.13204906$ |
0.586444209 |
\( \frac{30664297}{297} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 93 a - 537\) , \( -999 a + 7128\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(93a-537\right){x}-999a+7128$ |
| 33.1-c4 |
33.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.883253066$ |
0.586444209 |
\( \frac{347873904937}{395307} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1913 a - 13137\) , \( 115761 a - 801162\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1913a-13137\right){x}+115761a-801162$ |
| 33.1-d1 |
33.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{36} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.631790748$ |
$2.234063206$ |
5.274338318 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -5093 a + 35268\) , \( -193596 a + 1340204\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5093a+35268\right){x}-193596a+1340204$ |
| 33.1-d2 |
33.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{24} \cdot 11^{4} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.631790748$ |
$8.936252827$ |
5.274338318 |
\( \frac{169112377}{88209} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 1342 a - 9282\) , \( -14406 a + 99734\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1342a-9282\right){x}-14406a+99734$ |
| 33.1-d3 |
33.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{18} \cdot 11^{2} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.631790748$ |
$8.936252827$ |
5.274338318 |
\( \frac{30664297}{297} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 757 a - 5232\) , \( 32808 a - 227110\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(757a-5232\right){x}+32808a-227110$ |
| 33.1-d4 |
33.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{18} \cdot 11^{8} \) |
$2.75112$ |
$(3,a+1), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.157947687$ |
$8.936252827$ |
5.274338318 |
\( \frac{347873904937}{395307} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 17137 a - 118632\) , \( -3003792 a + 20794100\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(17137a-118632\right){x}-3003792a+20794100$ |
*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.
