| 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 |
| 33.2-a1 |
33.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{10} \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 6 a - 5\) , \( -3 a + 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-5\right){x}-3a+8$ |
| 33.2-a2 |
33.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{2} \cdot 11^{5} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 460938 a - 798369\) , \( -223745835 a + 387539154\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(460938a-798369\right){x}-223745835a+387539154$ |
| 33.2-a3 |
33.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{10} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 8273 a - 14331\) , \( -535987 a + 928356\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(8273a-14331\right){x}-535987a+928356$ |
| 33.2-a4 |
33.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{5} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( -7 a - 8\) , \( -12 a - 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-8\right){x}-12a-19$ |
| 33.2-b1 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( -\frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+{x}$ |
| 33.2-b2 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{8} \cdot 11^{8} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$3.864064054$ |
1.115459211 |
\( \frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 25 a - 29\) , \( -89 a + 138\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(25a-29\right){x}-89a+138$ |
| 33.2-b3 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{4} \cdot 11^{4} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( -\frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 20 a - 39\) , \( -84 a + 138\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(20a-39\right){x}-84a+138$ |
| 33.2-b4 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( -\frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 335 a - 609\) , \( -4719 a + 8214\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(335a-609\right){x}-4719a+8214$ |
| 33.2-b5 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( \frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -4\) , \( -4 a - 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}-4{x}-4a-1$ |
| 33.2-b6 |
33.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$3.864064054$ |
1.115459211 |
\( \frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -20 a - 49\) , \( -100 a - 184\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-20a-49\right){x}-100a-184$ |
| 33.2-c1 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( -\frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$ |
| 33.2-c2 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{8} \cdot 11^{8} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.663525979$ |
0.480218586 |
\( \frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 25 a - 29\) , \( 89 a - 138\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(25a-29\right){x}+89a-138$ |
| 33.2-c3 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{4} \cdot 11^{4} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$6.654103918$ |
0.480218586 |
\( -\frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 20 a - 39\) , \( 84 a - 138\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(20a-39\right){x}+84a-138$ |
| 33.2-c4 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.663525979$ |
0.480218586 |
\( -\frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 335 a - 609\) , \( 4719 a - 8214\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(335a-609\right){x}+4719a-8214$ |
| 33.2-c5 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( \frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( 4 a + 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}+4a+1$ |
| 33.2-c6 |
33.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( \frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -20 a - 49\) , \( 100 a + 184\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-20a-49\right){x}+100a+184$ |
| 33.2-d1 |
33.2-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{10} \cdot 11 \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$18.15775980$ |
0.524169375 |
\( -\frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 5 a - 7\) , \( a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5a-7\right){x}+a-1$ |
| 33.2-d2 |
33.2-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{2} \cdot 11^{5} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.726310392$ |
0.524169375 |
\( -\frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 460939 a - 798370\) , \( 224206774 a - 388337524\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(460939a-798370\right){x}+224206774a-388337524$ |
| 33.2-d3 |
33.2-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{10} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.726310392$ |
0.524169375 |
\( \frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 8272 a - 14332\) , \( 544260 a - 942688\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(8272a-14332\right){x}+544260a-942688$ |
| 33.2-d4 |
33.2-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.2 |
\( 3 \cdot 11 \) |
\( - 3^{5} \cdot 11^{2} \) |
$0.74192$ |
$(a), (-2a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$18.15775980$ |
0.524169375 |
\( \frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -6 a - 9\) , \( 6 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-6a-9\right){x}+6a+10$ |
*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.
