| 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 |
| 676.1-a1 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{8} \cdot 13^{11} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.558447599$ |
$2.900707737$ |
4.812906061 |
\( \frac{62097103570327}{169671989968} a + \frac{68165615175361}{84835994984} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -117 a - 254\) , \( -533 a - 1161\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-117a-254\right){x}-533a-1161$ |
| 676.1-a2 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{24} \cdot 13^{9} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$1.675342797$ |
$0.322300859$ |
4.812906061 |
\( \frac{1899143322426205933}{9885304832} a + \frac{8328056440432901151}{19770609664} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 814 a - 2651\) , \( 856772 a - 2735559\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(814a-2651\right){x}+856772a-2735559$ |
| 676.1-a3 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{8} \cdot 13^{3} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2^{3} \) |
$5.026028391$ |
$0.035811206$ |
4.812906061 |
\( \frac{556040942679200429007305410291}{2704} a + \frac{1219165586581361880199178542451}{2704} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 414734 a - 1328651\) , \( 239332100 a - 764213575\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(414734a-1328651\right){x}+239332100a-764213575$ |
| 676.1-b1 |
676.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{6} \cdot 13^{8} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$2.000052427$ |
$0.586665095$ |
3.921977848 |
\( -\frac{121322804009960125}{19307236} a + \frac{774665353930383375}{38614472} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 1105 a - 3549\) , \( 32272 a - 103105\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1105a-3549\right){x}+32272a-103105$ |
| 676.1-b2 |
676.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{2} \cdot 13^{8} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$6.000157282$ |
$5.279985862$ |
3.921977848 |
\( -\frac{4255077625}{9653618} a + \frac{14842499375}{4826809} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 15 a - 44\) , \( 29 a - 92\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(15a-44\right){x}+29a-92$ |
| 676.1-b3 |
676.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{4} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.000078641$ |
$21.11994345$ |
3.921977848 |
\( -\frac{26868375}{8788} a + \frac{101510375}{8788} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 5 a - 14\) , \( -13 a + 40\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5a-14\right){x}-13a+40$ |
| 676.1-b4 |
676.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 13^{4} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1.000026213$ |
$2.346660383$ |
3.921977848 |
\( \frac{1306937550375}{17576} a + \frac{22950516448625}{140608} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 65 a - 229\) , \( 456 a - 1529\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(65a-229\right){x}+456a-1529$ |
| 676.1-c1 |
676.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{8} \cdot 13^{3} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2^{3} \) |
$5.026028391$ |
$0.035811206$ |
4.812906061 |
\( -\frac{556040942679200429007305410291}{2704} a + \frac{887603264630281154603241976371}{1352} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -414735 a - 913916\) , \( -239332100 a - 524881475\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-414735a-913916\right){x}-239332100a-524881475$ |
| 676.1-c2 |
676.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{24} \cdot 13^{9} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$1.675342797$ |
$0.322300859$ |
4.812906061 |
\( -\frac{1899143322426205933}{9885304832} a + \frac{12126343085285313017}{19770609664} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -815 a - 1836\) , \( -856772 a - 1878787\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-815a-1836\right){x}-856772a-1878787$ |
| 676.1-c3 |
676.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{8} \cdot 13^{11} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.558447599$ |
$2.900707737$ |
4.812906061 |
\( -\frac{62097103570327}{169671989968} a + \frac{198428333921049}{169671989968} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 116 a - 370\) , \( 532 a - 1693\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(116a-370\right){x}+532a-1693$ |
| 676.1-d1 |
676.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{4} \cdot 13^{3} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.038601227$ |
$29.40148540$ |
1.686014759 |
\( -\frac{75978547}{676} a + \frac{241853925}{676} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -a - 3\) , \( a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-a-3\right){x}+a+3$ |
| 676.1-e1 |
676.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{2} \cdot 13^{8} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$6.000157282$ |
$5.279985862$ |
3.921977848 |
\( \frac{4255077625}{9653618} a + \frac{25429921125}{9653618} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -16 a - 29\) , \( -30 a - 63\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16a-29\right){x}-30a-63$ |
| 676.1-e2 |
676.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{4} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.000078641$ |
$21.11994345$ |
3.921977848 |
\( \frac{26868375}{8788} a + \frac{18660500}{2197} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -6 a - 9\) , \( 12 a + 27\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a-9\right){x}+12a+27$ |
| 676.1-e3 |
676.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 13^{4} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1.000026213$ |
$2.346660383$ |
3.921977848 |
\( -\frac{1306937550375}{17576} a + \frac{33406016851625}{140608} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -66 a - 164\) , \( -457 a - 1073\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-66a-164\right){x}-457a-1073$ |
| 676.1-e4 |
676.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{6} \cdot 13^{8} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$2.000052427$ |
$0.586665095$ |
3.921977848 |
\( \frac{121322804009960125}{19307236} a + \frac{532019745910463125}{38614472} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -1106 a - 2444\) , \( -32273 a - 70833\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1106a-2444\right){x}-32273a-70833$ |
| 676.1-f1 |
676.1-f |
$2$ |
$7$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$29.18461737$ |
$0.385597965$ |
4.179455775 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$ |
| 676.1-f2 |
676.1-f |
$2$ |
$7$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$4.169231053$ |
$18.89430030$ |
4.179455775 |
\( -\frac{2146689}{1664} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 676.1-g1 |
676.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$2.236970459$ |
$0.265819283$ |
1.987559954 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$ |
| 676.1-g2 |
676.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$0.745656819$ |
$2.392373550$ |
1.987559954 |
\( -\frac{10218313}{17576} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5{x}-8$ |
| 676.1-g3 |
676.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$2.236970459$ |
$21.53136195$ |
1.987559954 |
\( \frac{12167}{26} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}$ |
| 676.1-h1 |
676.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( - 2^{4} \cdot 13^{3} \) |
$2.45372$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.038601227$ |
$29.40148540$ |
1.686014759 |
\( \frac{75978547}{676} a + \frac{82937689}{338} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -3\) , \( -a + 4\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}-3{x}-a+4$ |
*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.
