| 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 |
| 27225.4-a1 |
27225.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{8} \cdot 11^{3} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.763924145$ |
$1.384648879$ |
5.102858613 |
\( -\frac{9137}{2025} a - \frac{74746}{2025} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 3 a - 5\) , \( 30 a - 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(3a-5\right){x}+30a-21$ |
| 27225.4-a2 |
27225.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{14} \cdot 5^{7} \cdot 11^{3} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.527848290$ |
$0.692324439$ |
5.102858613 |
\( -\frac{507753467}{32805} a + \frac{796676879}{32805} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -87 a - 70\) , \( 496 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-87a-70\right){x}+496a+10$ |
| 27225.4-b1 |
27225.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{5} \cdot 5^{8} \cdot 11^{9} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.437242627$ |
$0.485407095$ |
6.143325203 |
\( -\frac{2854912}{3267} a + \frac{4784128}{3267} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 33 a - 205\) , \( 486 a - 762\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(33a-205\right){x}+486a-762$ |
| 27225.4-b2 |
27225.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{8} \cdot 11^{7} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{3} \) |
$1.311727881$ |
$0.485407095$ |
6.143325203 |
\( -\frac{305475584}{8019} a + \frac{137142272}{2673} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -187 a + 400\) , \( -702 a - 2995\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-187a+400\right){x}-702a-2995$ |
| 27225.4-c1 |
27225.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{5} \cdot 5^{2} \cdot 11^{9} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.545155217$ |
$1.085403261$ |
2.854532127 |
\( -\frac{2854912}{3267} a + \frac{4784128}{3267} \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( 23 a + 3\) , \( -34 a + 116\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(23a+3\right){x}-34a+116$ |
| 27225.4-c2 |
27225.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{2} \cdot 11^{7} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.181718405$ |
$1.085403261$ |
2.854532127 |
\( -\frac{305475584}{8019} a + \frac{137142272}{2673} \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -32 a - 52\) , \( 197 a + 72\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-32a-52\right){x}+197a+72$ |
| 27225.4-d1 |
27225.4-d |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{4} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.107600940$ |
$1.596960189$ |
3.730321925 |
\( \frac{45056}{27} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 12 a - 8\) , \( -3 a + 8\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(12a-8\right){x}-3a+8$ |
| 27225.4-e1 |
27225.4-e |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 5^{6} \cdot 11^{2} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.861764789$ |
$1.917754005$ |
7.972697452 |
\( \frac{1068415}{729} a + \frac{709403}{729} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 3 a + 11\) , \( -6 a + 23\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a+11\right){x}-6a+23$ |
| 27225.4-e2 |
27225.4-e |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 5^{6} \cdot 11^{2} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.287254929$ |
$1.917754005$ |
7.972697452 |
\( -\frac{1068415}{729} a + \frac{592606}{243} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 2 a - 16\) , \( -4 a + 14\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-16\right){x}-4a+14$ |
| 27225.4-f1 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{12} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{6} \cdot 5 \) |
$0.284422680$ |
$0.534743389$ |
3.668624767 |
\( \frac{349209575}{59049} a - \frac{1245002978}{59049} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 166 a - 142\) , \( 1009 a + 71\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(166a-142\right){x}+1009a+71$ |
| 27225.4-f2 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{12} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{6} \) |
$1.422113402$ |
$0.534743389$ |
3.668624767 |
\( -\frac{349209575}{59049} a - \frac{298597801}{19683} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 171 a - 86\) , \( -750 a - 1149\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(171a-86\right){x}-750a-1149$ |
| 27225.4-f3 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.568845360$ |
$1.069486778$ |
3.668624767 |
\( \frac{77935}{243} a - \frac{112717}{243} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( a - 32\) , \( -3 a + 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(a-32\right){x}-3a+104$ |
| 27225.4-f4 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{21} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \) |
$2.844226804$ |
$0.267371694$ |
3.668624767 |
\( \frac{2927543402641}{3486784401} a - \frac{430814699872}{1162261467} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 226 a - 581\) , \( 2143 a - 7386\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(226a-581\right){x}+2143a-7386$ |
| 27225.4-f5 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.844226804$ |
$1.069486778$ |
3.668624767 |
\( -\frac{77935}{243} a - \frac{11594}{81} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 6 a + 24\) , \( -57 a + 39\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(6a+24\right){x}-57a+39$ |
| 27225.4-f6 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{21} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.568845360$ |
$0.267371694$ |
3.668624767 |
\( -\frac{2927543402641}{3486784401} a + \frac{1635099303025}{3486784401} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 276 a + 243\) , \( 3044 a - 7244\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(276a+243\right){x}+3044a-7244$ |
| 27225.4-f7 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \) |
$2.844226804$ |
$0.267371694$ |
3.668624767 |
\( \frac{54238838797}{243} a + \frac{1331574640}{9} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 2756 a - 1351\) , \( -49535 a - 62584\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(2756a-1351\right){x}-49535a-62584$ |
| 27225.4-f8 |
27225.4-f |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{6} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.568845360$ |
$0.267371694$ |
3.668624767 |
\( -\frac{54238838797}{243} a + \frac{90191354077}{243} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 2696 a - 2287\) , \( 63522 a + 18254\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(2696a-2287\right){x}+63522a+18254$ |
| 27225.4-g1 |
27225.4-g |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.4 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{8} \cdot 11^{8} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.548674873$ |
3.308633975 |
\( -\frac{2449408}{6075} a + \frac{2240512}{2025} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 55 a - 128\) , \( 341 a + 50\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(55a-128\right){x}+341a+50$ |
*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.
