| 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-a6 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$34.81892277$ |
0.336733136 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 746 a - 2517\) , \( -18744 a + 63210\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(746a-2517\right){x}-18744a+63210$ |
| 4.1-b6 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$1.489742545$ |
1.166989006 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -293 a - 696\) , \( -5104 a - 12109\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-293a-696\right){x}-5104a-12109$ |
| 128.5-c6 |
128.5-c |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$2.546351081$ |
3.989365447 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3343 a - 7940\) , \( -177777 a - 421736\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3343a-7940\right){x}-177777a-421736$ |
| 128.5-g6 |
128.5-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.504596085$ |
$2.546351081$ |
2.667726059 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 4198 a - 14161\) , \( 254442 a - 858049\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(4198a-14161\right){x}+254442a-858049$ |
| 128.6-c6 |
128.6-c |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$2.546351081$ |
3.989365447 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -1647 a - 3929\) , \( 57605 a + 136627\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-1647a-3929\right){x}+57605a+136627$ |
| 128.6-g6 |
128.6-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.72662$ |
$(-a-2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.009192170$ |
$2.546351081$ |
2.667726059 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 8484 a - 28613\) , \( -710360 a + 2395527\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8484a-28613\right){x}-710360a+2395527$ |
| 256.1-e6 |
256.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{27} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$8.704730694$ |
3.030598229 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4699 a - 11176\) , \( 301374 a + 714980\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4699a-11176\right){x}+301374a+714980$ |
| 256.1-j6 |
256.1-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{27} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{4} \) |
$1$ |
$0.372435636$ |
2.333978012 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 11936 a - 40264\) , \( 1199616 a - 4045456\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(11936a-40264\right){x}+1199616a-4045456$ |
| 288.3-g6 |
288.3-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.430051629$ |
1.347522833 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -15979 a - 37912\) , \( -1873391 a - 4444216\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-15979a-37912\right){x}-1873391a-4444216$ |
| 288.3-j6 |
288.3-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.338294176$ |
$10.05135721$ |
4.735351769 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1969 a - 6688\) , \( -80821 a + 272632\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1969a-6688\right){x}-80821a+272632$ |
| 288.4-g6 |
288.4-g |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.430051629$ |
1.347522833 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -760 a - 1912\) , \( -20074 a - 48134\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-760a-1912\right){x}-20074a-48134$ |
| 288.4-j6 |
288.4-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.676588352$ |
$10.05135721$ |
4.735351769 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 40551 a - 136761\) , \( -7510500 a + 25327515\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(40551a-136761\right){x}-7510500a+25327515$ |
| 484.1-a6 |
484.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
484.1 |
\( 2^{2} \cdot 11^{2} \) |
\( - 2^{3} \cdot 11^{6} \) |
$2.40772$ |
$(-a-2), (-a+3), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$2.171535498$ |
3.402142284 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 108 a - 766\) , \( 2498 a - 5650\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(108a-766\right){x}+2498a-5650$ |
| 484.1-j6 |
484.1-j |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
484.1 |
\( 2^{2} \cdot 11^{2} \) |
\( - 2^{3} \cdot 11^{6} \) |
$2.40772$ |
$(-a-2), (-a+3), (-4a-9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$2.171535498$ |
3.402142284 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -148704 a - 352767\) , \( -52962336 a - 125641561\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-148704a-352767\right){x}-52962336a-125641561$ |
*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.
