| 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 |
| 33.1-a1 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{10} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 22986 a - 39809\) , \( -2497992 a + 4326651\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22986a-39809\right){x}-2497992a+4326651$ |
| 33.1-a2 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{5} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 6 a - 8\) , \( 12 a - 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(6a-8\right){x}+12a-19$ |
| 33.1-a3 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{10} \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 15 a - 28\) , \( 96 a - 167\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-28\right){x}+96a-167$ |
| 33.1-a4 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{2} \cdot 11^{5} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 57891 a - 100269\) , \( -25338895 a + 43888254\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(57891a-100269\right){x}-25338895a+43888254$ |
| 33.1-b1 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{8} \cdot 11^{8} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$3.864064054$ |
1.115459211 |
\( -\frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -727 a + 1262\) , \( 19515 a - 33800\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-727a+1262\right){x}+19515a-33800$ |
| 33.1-b2 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( \frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 3362 a - 5824\) , \( 177111 a - 306766\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(3362a-5824\right){x}+177111a-306766$ |
| 33.1-b3 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$3.864064054$ |
1.115459211 |
\( -\frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 20 a - 49\) , \( 100 a - 184\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(20a-49\right){x}+100a-184$ |
| 33.1-b4 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( -\frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 298 a - 513\) , \( 3675 a - 6364\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(298a-513\right){x}+3675a-6364$ |
| 33.1-b5 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{4} \cdot 11^{4} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( \frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 318 a - 548\) , \( 3150 a - 5455\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(318a-548\right){x}+3150a-5455$ |
| 33.1-b6 |
33.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.45625621$ |
1.115459211 |
\( \frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 1683 a - 2918\) , \( -46815 a + 81086\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1683a-2918\right){x}-46815a+81086$ |
| 33.1-c1 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{8} \cdot 11^{8} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.663525979$ |
0.480218586 |
\( -\frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -729 a + 1261\) , \( -20243 a + 35060\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-729a+1261\right){x}-20243a+35060$ |
| 33.1-c2 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( \frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 3362 a - 5824\) , \( -177111 a + 306765\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(3362a-5824\right){x}-177111a+306765$ |
| 33.1-c3 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( -\frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 20 a - 49\) , \( -100 a + 184\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(20a-49\right){x}-100a+184$ |
| 33.1-c4 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$26.61641567$ |
0.480218586 |
\( -\frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 296 a - 514\) , \( -3378 a + 5849\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(296a-514\right){x}-3378a+5849$ |
| 33.1-c5 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{4} \cdot 11^{4} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$6.654103918$ |
0.480218586 |
\( \frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 316 a - 549\) , \( -2833 a + 4905\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(316a-549\right){x}-2833a+4905$ |
| 33.1-c6 |
33.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{2} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.663525979$ |
0.480218586 |
\( \frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 1681 a - 2919\) , \( 48497 a - 84006\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1681a-2919\right){x}+48497a-84006$ |
| 33.1-d1 |
33.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{10} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.726310392$ |
0.524169375 |
\( -\frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 22985 a - 39809\) , \( 2481167 a - 4297507\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(22985a-39809\right){x}+2481167a-4297507$ |
| 33.1-d2 |
33.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{5} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$18.15775980$ |
0.524169375 |
\( -\frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 5 a - 9\) , \( -6 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(5a-9\right){x}-6a+10$ |
| 33.1-d3 |
33.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{10} \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$18.15775980$ |
0.524169375 |
\( \frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 16 a - 29\) , \( -80 a + 138\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(16a-29\right){x}-80a+138$ |
| 33.1-d4 |
33.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{2} \cdot 11^{5} \) |
$0.74192$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.726310392$ |
0.524169375 |
\( \frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 57890 a - 100270\) , \( 25396786 a - 43988524\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(57890a-100270\right){x}+25396786a-43988524$ |
*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.
