| 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 |
| 28.1-a1 |
28.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{7} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.446647524$ |
$12.17207760$ |
4.694109082 |
\( \frac{209009}{1568} a + \frac{76935}{98} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -183 a + 1267\) , \( -57821 a + 395733\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-183a+1267\right){x}-57821a+395733$ |
| 28.1-a2 |
28.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{11} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.223323762$ |
$24.34415521$ |
4.694109082 |
\( -\frac{15590041}{7168} a + \frac{14913995}{896} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 10\) , \( -a - 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+10\right){x}-a-5$ |
| 28.1-b1 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$17.01778343$ |
$0.436190660$ |
2.340056850 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 28.1-b2 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$1.890864825$ |
$35.33144352$ |
2.340056850 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 28.1-b3 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$5.672594476$ |
$3.925715946$ |
2.340056850 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 28.1-b4 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$11.34518895$ |
$3.925715946$ |
2.340056850 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 28.1-b5 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$3.781729651$ |
$35.33144352$ |
2.340056850 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 28.1-b6 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$34.03556686$ |
$0.436190660$ |
2.340056850 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
| 28.1-c1 |
28.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{7} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.446647524$ |
$12.17207760$ |
4.694109082 |
\( -\frac{209009}{1568} a + \frac{1439969}{1568} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 183 a + 1084\) , \( 57821 a + 337912\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(183a+1084\right){x}+57821a+337912$ |
| 28.1-c2 |
28.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{11} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.223323762$ |
$24.34415521$ |
4.694109082 |
\( \frac{15590041}{7168} a + \frac{14817417}{1024} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a + 11\) , \( a + 5\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+11\right){x}+a+5$ |
| 28.1-d1 |
28.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{7} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$7.720651804$ |
1.216945205 |
\( -\frac{209009}{1568} a + \frac{1439969}{1568} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -11957 a + 81859\) , \( -222525 a + 1523045\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-11957a+81859\right){x}-222525a+1523045$ |
| 28.1-d2 |
28.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{11} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.44130360$ |
1.216945205 |
\( \frac{15590041}{7168} a + \frac{14817417}{1024} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 82177726 a - 562448068\) , \( -135663600270 a + 928520855840\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(82177726a-562448068\right){x}-135663600270a+928520855840$ |
| 28.1-e1 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.697542992$ |
$7.027708105$ |
3.090734813 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -7453261000 a - 43559009551\) , \( 884743726381089 a + 5170697824615794\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-7453261000a-43559009551\right){x}+884743726381089a+5170697824615794$ |
| 28.1-e2 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.697542992$ |
$7.027708105$ |
3.090734813 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -22765000 a - 133045221\) , \( -301470021617 a - 1761877861908\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-22765000a-133045221\right){x}-301470021617a-1761877861908$ |
| 28.1-e3 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$0.697542992$ |
$7.027708105$ |
3.090734813 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -195779001 a + 1339968025\) , \( -6303992811223 a + 43146347205413\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-195779001a+1339968025\right){x}-6303992811223a+43146347205413$ |
| 28.1-e4 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2 \) |
$2.790171969$ |
$7.027708105$ |
3.090734813 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 1552572999 a - 10626257935\) , \( -68746620820639 a + 470521724873581\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1552572999a-10626257935\right){x}-68746620820639a+470521724873581$ |
| 28.1-e5 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$2.790171969$ |
$7.027708105$ |
3.090734813 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 459852999 a - 3147366710\) , \( 13512395687294 a - 92482738061398\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(459852999a-3147366710\right){x}+13512395687294a-92482738061398$ |
| 28.1-e6 |
28.1-e |
$6$ |
$18$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$2.790171969$ |
$7.027708105$ |
3.090734813 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -119347789000 a - 697502942991\) , \( 56529348060097057 a + 330373834055752050\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-119347789000a-697502942991\right){x}+56529348060097057a+330373834055752050$ |
| 28.1-f1 |
28.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{7} \cdot 7^{4} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$7.720651804$ |
1.216945205 |
\( \frac{209009}{1568} a + \frac{76935}{98} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 11959 a + 69900\) , \( 210567 a + 1230619\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11959a+69900\right){x}+210567a+1230619$ |
| 28.1-f2 |
28.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{161}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{11} \cdot 7^{2} \) |
$2.60820$ |
$(a+6), (-a+7), (-32a+219)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.44130360$ |
1.216945205 |
\( -\frac{15590041}{7168} a + \frac{14913995}{896} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -82177724 a - 480270343\) , \( 135581422545 a + 792376985227\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-82177724a-480270343\right){x}+135581422545a+792376985227$ |
*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.
