| 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 |
| 275.1-a1 |
275.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.645804326$ |
$4.539063627$ |
2.252413531 |
\( \frac{3825152}{3025} a - \frac{7868992}{3025} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 7 a + 22\) , \( 11 a + 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(7a+22\right){x}+11a+33$ |
| 275.1-a2 |
275.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.822902163$ |
$9.078127255$ |
2.252413531 |
\( -\frac{2340301312}{275} a + \frac{7762416448}{275} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 84 a - 259\) , \( 694 a - 2278\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(84a-259\right){x}+694a-2278$ |
| 275.1-b1 |
275.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.128463704$ |
$2.918250465$ |
2.938867532 |
\( -\frac{3574542501217453059}{13427734375} a - \frac{11855416306009927554}{13427734375} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 22 a - 109\) , \( 3885 a - 12801\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(22a-109\right){x}+3885a-12801$ |
| 275.1-c1 |
275.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.645804326$ |
$4.539063627$ |
2.252413531 |
\( -\frac{3825152}{3025} a - \frac{7868992}{3025} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -4 a + 27\) , \( 11 a - 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a+27\right){x}+11a-25$ |
| 275.1-c2 |
275.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.822902163$ |
$9.078127255$ |
2.252413531 |
\( \frac{2340301312}{275} a + \frac{7762416448}{275} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -77 a - 254\) , \( -953 a - 3161\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-77a-254\right){x}-953a-3161$ |
| 275.1-d1 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( - 5^{10} \cdot 11 \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$2.819393336$ |
$0.740695571$ |
2.518599226 |
\( -\frac{7203019491662585787}{4296875} a + \frac{2171792091488681448}{390625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 135 a - 500\) , \( 1826 a - 6344\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(135a-500\right){x}+1826a-6344$ |
| 275.1-d2 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$5.638786672$ |
$11.85112914$ |
2.518599226 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
| 275.1-d3 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.819393336$ |
$11.85112914$ |
2.518599226 |
\( \frac{8120601}{3025} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}-8$ |
| 275.1-d4 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{2} \cdot 11^{8} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.409696668$ |
$11.85112914$ |
2.518599226 |
\( \frac{2749884201}{73205} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -30\) , \( 22\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-30{x}+22$ |
| 275.1-d5 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{8} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.409696668$ |
$2.962782285$ |
2.518599226 |
\( \frac{22930509321}{6875} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -60\) , \( -250\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-60{x}-250$ |
| 275.1-d6 |
275.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( - 5^{10} \cdot 11 \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$2.819393336$ |
$0.740695571$ |
2.518599226 |
\( \frac{7203019491662585787}{4296875} a + \frac{2171792091488681448}{390625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -135 a - 500\) , \( -1826 a - 6344\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-135a-500\right){x}-1826a-6344$ |
| 275.1-e1 |
275.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$1$ |
$0.778762871$ |
3.052475924 |
\( \frac{3574542501217453059}{13427734375} a - \frac{11855416306009927554}{13427734375} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -23 a - 110\) , \( 3862 a + 12688\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-23a-110\right){x}+3862a+12688$ |
| 275.1-f1 |
275.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.128463704$ |
$2.918250465$ |
2.938867532 |
\( \frac{3574542501217453059}{13427734375} a - \frac{11855416306009927554}{13427734375} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -23 a - 109\) , \( -3885 a - 12801\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-23a-109\right){x}-3885a-12801$ |
| 275.1-g1 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( - 5^{10} \cdot 11 \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$8.465293384$ |
1.276190995 |
\( -\frac{7203019491662585787}{4296875} a + \frac{2171792091488681448}{390625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 135 a - 499\) , \( -1691 a + 5844\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(135a-499\right){x}-1691a+5844$ |
| 275.1-g2 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.93058676$ |
1.276190995 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+{x}$ |
| 275.1-g3 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$16.93058676$ |
1.276190995 |
\( \frac{8120601}{3025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -4\) , \( 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-4{x}+3$ |
| 275.1-g4 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{2} \cdot 11^{8} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$4.232646692$ |
1.276190995 |
\( \frac{2749884201}{73205} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -29\) , \( -52\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-29{x}-52$ |
| 275.1-g5 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{8} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$16.93058676$ |
1.276190995 |
\( \frac{22930509321}{6875} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -59\) , \( 190\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-59{x}+190$ |
| 275.1-g6 |
275.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( - 5^{10} \cdot 11 \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$8.465293384$ |
1.276190995 |
\( \frac{7203019491662585787}{4296875} a + \frac{2171792091488681448}{390625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -135 a - 499\) , \( 1691 a + 5844\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-135a-499\right){x}+1691a+5844$ |
| 275.1-h1 |
275.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.90088838$ |
4.191275546 |
\( -\frac{3825152}{3025} a - \frac{7868992}{3025} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -2 a + 31\) , \( 2 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+31\right){x}+2a+19$ |
| 275.1-h2 |
275.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.80177676$ |
4.191275546 |
\( \frac{2340301312}{275} a + \frac{7762416448}{275} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -79 a - 258\) , \( 531 a + 1760\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-79a-258\right){x}+531a+1760$ |
| 275.1-i1 |
275.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$1$ |
$0.778762871$ |
3.052475924 |
\( -\frac{3574542501217453059}{13427734375} a - \frac{11855416306009927554}{13427734375} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 22 a - 110\) , \( -3863 a + 12688\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(22a-110\right){x}-3863a+12688$ |
| 275.1-j1 |
275.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{4} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.90088838$ |
4.191275546 |
\( \frac{3825152}{3025} a - \frac{7868992}{3025} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 9 a + 26\) , \( 24 a + 77\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a+26\right){x}+24a+77$ |
| 275.1-j2 |
275.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
275.1 |
\( 5^{2} \cdot 11 \) |
\( 5^{3} \cdot 11^{2} \) |
$2.41378$ |
$(a-4), (-a-4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.80177676$ |
4.191275546 |
\( -\frac{2340301312}{275} a + \frac{7762416448}{275} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 82 a - 263\) , \( -794 a + 2643\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(82a-263\right){x}-794a+2643$ |
*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.
