| 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 |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5084708729 a - 41565845138\) , \( 582944485604320 a + 4765382171313953\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5084708729a-41565845138\right){x}+582944485604320a+4765382171313953$ |
| 28.1-a2 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2^{2} \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( -\frac{15625}{28} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 15530579 a - 142487838\) , \( 198672761675 a - 1822758175273\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(15530579a-142487838\right){x}+198672761675a-1822758175273$ |
| 28.1-a3 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \cdot 3 \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( \frac{9938375}{21952} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -133562896 a + 1225396457\) , \( -4153968052764 a + 38111310112069\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-133562896a+1225396457\right){x}-4153968052764a+38111310112069$ |
| 28.1-a4 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2 \cdot 3 \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 1059184904 a - 9717677903\) , \( -45294177781220 a + 415559396157157\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1059184904a-9717677903\right){x}-45294177781220a+415559396157157$ |
| 28.1-a5 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2 \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( \frac{128787625}{98} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 313717529 a - 2878256428\) , \( 8903954390553 a - 81690894749957\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(313717529a-2878256428\right){x}+8903954390553a-81690894749957$ |
| 28.1-a6 |
28.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$3.56625$ |
$(-23a+211), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2 \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
11.13831196 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 81420567929 a - 747007312908\) , \( -37246591460814439 a + 341725400803283189\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(81420567929a-747007312908\right){x}-37246591460814439a+341725400803283189$ |
| 28.1-b1 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.436190660$ |
0.905098021 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 28.1-b2 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \) |
$1$ |
$35.33144352$ |
0.905098021 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 28.1-b3 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.925715946$ |
0.905098021 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 28.1-b4 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.925715946$ |
0.905098021 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 28.1-b5 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \) |
$1$ |
$35.33144352$ |
0.905098021 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 28.1-b6 |
28.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{301}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$3.56625$ |
$(-23a+211), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.436190660$ |
0.905098021 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
*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.
