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 |
539.1-a1 |
539.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$0.602881548$ |
1.679171307 |
\( -\frac{78843215872}{539} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 32875 a - 110860\) , \( 5524839 a - 18631313\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(32875a-110860\right){x}+5524839a-18631313$ |
539.1-a2 |
539.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{12} \cdot 11^{6} \) |
$2.47339$ |
$(-4a-9), (7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.602881548$ |
1.679171307 |
\( -\frac{13278380032}{156590819} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -18155 a - 43065\) , \( -10448199 a - 24786069\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-18155a-43065\right){x}-10448199a-24786069$ |
539.1-a3 |
539.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{18} \) |
$2.47339$ |
$(-4a-9), (7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.602881548$ |
1.679171307 |
\( \frac{9463555063808}{115539436859} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -162165 a + 546870\) , \( -270199791 a + 911189707\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-162165a+546870\right){x}-270199791a+911189707$ |
539.1-b1 |
539.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{6} \cdot 11^{4} \) |
$2.47339$ |
$(-4a-9), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.031061357$ |
3.671852040 |
\( \frac{4657463}{41503} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 4\) , \( 11\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+4{x}+11$ |
539.1-b2 |
539.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{12} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.031061357$ |
3.671852040 |
\( \frac{15124197817}{1294139} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -51\) , \( 110\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-51{x}+110$ |
539.1-c1 |
539.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.098027979$ |
$10.23860076$ |
1.397731252 |
\( \frac{884736}{539} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 2\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+2{x}$ |
539.1-d1 |
539.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.099137699$ |
$9.086092621$ |
2.508874914 |
\( \frac{884736}{539} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 736 a + 1746\) , \( -4230 a - 10035\bigr] \) |
${y}^2+{y}={x}^{3}+\left(736a+1746\right){x}-4230a-10035$ |
539.1-e1 |
539.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{6} \cdot 11^{4} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.183928702$ |
$3.210187340$ |
1.984815816 |
\( \frac{4657463}{41503} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 1280 a + 3040\) , \( -161701 a - 383601\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1280a+3040\right){x}-161701a-383601$ |
539.1-e2 |
539.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{12} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.591964351$ |
$3.210187340$ |
1.984815816 |
\( \frac{15124197817}{1294139} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -18960 a - 44975\) , \( -2203466 a - 5227242\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-18960a-44975\right){x}-2203466a-5227242$ |
539.1-f1 |
539.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{2} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.745116266$ |
$21.59210152$ |
2.489484723 |
\( -\frac{78843215872}{539} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -89\) , \( 295\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-89{x}+295$ |
539.1-f2 |
539.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{12} \cdot 11^{6} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.235348799$ |
$2.399122391$ |
2.489484723 |
\( -\frac{13278380032}{156590819} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -49\) , \( 600\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-49{x}+600$ |
539.1-f3 |
539.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
539.1 |
\( 7^{2} \cdot 11 \) |
\( 7^{4} \cdot 11^{18} \) |
$2.47339$ |
$(-4a-9), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \) |
$6.706046397$ |
$0.266569154$ |
2.489484723 |
\( \frac{9463555063808}{115539436859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 441\) , \( -15815\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+441{x}-15815$ |
*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.