| 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 |
| 8379.5-a1 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{12} \cdot 7^{7} \cdot 19^{8} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.135600973$ |
1.252628137 |
\( \frac{1389689543960222201}{3209893414749} a - \frac{822796453232303930}{1069964471583} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -7912 a + 8171\) , \( -10322 a - 273673\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-7912a+8171\right){x}-10322a-273673$ |
| 8379.5-a2 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{30} \cdot 7^{7} \cdot 19^{2} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.135600973$ |
1.252628137 |
\( -\frac{111301988183011}{1342951407} a - \frac{2461906908050}{447650469} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -6532 a + 2141\) , \( -171056 a + 165017\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-6532a+2141\right){x}-171056a+165017$ |
| 8379.5-a3 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{18} \cdot 7^{8} \cdot 19^{4} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.271201947$ |
1.252628137 |
\( \frac{2872067964853}{4655196441} a + \frac{3343433306723}{4655196441} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -622 a + 476\) , \( -2951 a - 2080\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-622a+476\right){x}-2951a-2080$ |
| 8379.5-a4 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{12} \cdot 7^{10} \cdot 19^{2} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.542403894$ |
1.252628137 |
\( -\frac{28971353771}{23402547} a + \frac{23274207584}{7800849} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 203 a - 109\) , \( -452 a - 316\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(203a-109\right){x}-452a-316$ |
| 8379.5-a5 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{9} \cdot 7^{8} \cdot 19 \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.084807788$ |
1.252628137 |
\( \frac{238255387}{8379} a + \frac{59095573}{8379} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 83 a - 64\) , \( 241 a - 43\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(83a-64\right){x}+241a-43$ |
| 8379.5-a6 |
8379.5-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{9} \cdot 7^{14} \cdot 19 \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.271201947$ |
1.252628137 |
\( -\frac{3983977737373759}{985780971} a + \frac{3039882386782439}{985780971} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 2948 a - 1414\) , \( -40205 a - 16444\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(2948a-1414\right){x}-40205a-16444$ |
| 8379.5-b1 |
8379.5-b |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{3} \cdot 7^{2} \cdot 19 \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.079392041$ |
$6.010017162$ |
2.203850040 |
\( -\frac{771}{19} a - \frac{501}{19} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( a - 1\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a-1\right){x}-1$ |
| 8379.5-c1 |
8379.5-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{9} \cdot 7^{8} \cdot 19 \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.128215929$ |
$1.311493265$ |
2.330014716 |
\( -\frac{771}{19} a - \frac{501}{19} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3 a + 10\) , \( 51 a + 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-3a+10\right){x}+51a+12$ |
| 8379.5-d1 |
8379.5-d |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{3} \cdot 7^{6} \cdot 19^{3} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.776165441$ |
2.050939191 |
\( \frac{29840721}{6859} a - \frac{35267232}{6859} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 19 a - 17\) , \( 46 a - 16\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(19a-17\right){x}+46a-16$ |
| 8379.5-d2 |
8379.5-d |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{3} \cdot 7^{6} \cdot 19^{6} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.888082720$ |
2.050939191 |
\( -\frac{36038181633}{47045881} a - \frac{39546962313}{47045881} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -36 a - 27\) , \( 183 a + 49\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-36a-27\right){x}+183a+49$ |
| 8379.5-d3 |
8379.5-d |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{9} \cdot 7^{6} \cdot 19 \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.776165441$ |
2.050939191 |
\( -\frac{9153}{19} a + \frac{36801}{19} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 15 a\) , \( 12 a - 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+15a{x}+12a-16$ |
| 8379.5-d4 |
8379.5-d |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{9} \cdot 7^{6} \cdot 19^{2} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.888082720$ |
2.050939191 |
\( -\frac{363527109}{361} a + \frac{287391186}{361} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 180 a + 30\) , \( 297 a - 1207\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(180a+30\right){x}+297a-1207$ |
| 8379.5-e1 |
8379.5-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8379.5 |
\( 3^{2} \cdot 7^{2} \cdot 19 \) |
\( 3^{6} \cdot 7^{10} \cdot 19^{2} \) |
$1.48080$ |
$(-2a+1), (3a-2), (-5a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$1$ |
$0.960512948$ |
2.218209637 |
\( \frac{94208}{361} a + \frac{491520}{361} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( 17 a - 55\) , \( -53 a - 49\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a-55\right){x}-53a-49$ |
*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.
