| 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 |
| 42592.18-a1 |
42592.18-a |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{13} \cdot 11^{13} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$1$ |
$0.475624776$ |
1.078615607 |
\( \frac{96640575679}{82458112} a + \frac{5297923099}{82458112} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 64 a + 150\) , \( 351 a - 1190\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(64a+150\right){x}+351a-1190$ |
| 42592.18-b1 |
42592.18-b |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{11} \cdot 11^{3} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.115724623$ |
$4.003342475$ |
5.603372318 |
\( \frac{831}{22} a + \frac{34003}{22} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2 a - 2\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2a-2\right){x}$ |
| 42592.18-c1 |
42592.18-c |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{51} \cdot 11^{3} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1.469814848$ |
$0.399286256$ |
3.549097701 |
\( \frac{4070378798731}{11811160064} a + \frac{106097102447}{5905580032} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -148 a + 49\) , \( -135 a + 1967\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-148a+49\right){x}-135a+1967$ |
| 42592.18-c2 |
42592.18-c |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{25} \cdot 11^{5} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.489938282$ |
$1.197858769$ |
3.549097701 |
\( -\frac{531165637}{1362944} a + \frac{64260511}{681472} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 17 a - 6\) , \( -14 a - 68\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(17a-6\right){x}-14a-68$ |
| 42592.18-d1 |
42592.18-d |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{37} \cdot 11^{5} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$0.593634769$ |
1.346237115 |
\( -\frac{7726678638322513}{11433202941952} a + \frac{1801062937275851}{11433202941952} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -9 a + 111\) , \( -406 a - 159\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a+111\right){x}-406a-159$ |
| 42592.18-d2 |
42592.18-d |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{15} \cdot 11^{3} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$1.780904307$ |
1.346237115 |
\( -\frac{57280068041}{22528} a + \frac{30932642419}{22528} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 36 a + 16\) , \( 23 a - 170\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(36a+16\right){x}+23a-170$ |
| 42592.18-e1 |
42592.18-e |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{11} \cdot 11^{5} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1.060244760$ |
$2.607093527$ |
4.179012956 |
\( \frac{2463287}{1408} a - \frac{329101}{1408} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 4 a - 4\) , \( 5 a - 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(4a-4\right){x}+5a-4$ |
| 42592.18-f1 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{22} \cdot 11^{9} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.253403434$ |
2.298659893 |
\( -\frac{2775668240489}{85184} a - \frac{5083125111585}{85184} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 1419 a + 3140\) , \( -60410 a + 98640\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1419a+3140\right){x}-60410a+98640$ |
| 42592.18-f2 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{19} \cdot 11^{18} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$0.126701717$ |
2.298659893 |
\( \frac{41728910180660407}{200859416110144} a - \frac{51818768961625453}{200859416110144} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -356 a + 1804\) , \( -48952 a + 27540\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-356a+1804\right){x}-48952a+27540$ |
| 42592.18-f3 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{37} \cdot 11^{9} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.253403434$ |
2.298659893 |
\( \frac{74168468086089}{22330474496} a + \frac{16633019348749}{22330474496} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -469 a - 464\) , \( 7420 a - 788\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-469a-464\right){x}+7420a-788$ |
| 42592.18-f4 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{26} \cdot 11^{7} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.760210303$ |
2.298659893 |
\( -\frac{2222449}{45056} a + \frac{22133027}{22528} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 4 a + 65\) , \( 61 a + 195\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+65\right){x}+61a+195$ |
| 42592.18-f5 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{22} \cdot 11^{8} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.760210303$ |
2.298659893 |
\( \frac{998361}{7744} a + \frac{11225095}{3872} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -71 a + 29\) , \( -142 a + 294\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-71a+29\right){x}-142a+294$ |
| 42592.18-f6 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{26} \cdot 11^{12} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$1$ |
$0.253403434$ |
2.298659893 |
\( -\frac{49453830610989}{7256313856} a + \frac{47470228116479}{7256313856} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 764 a - 756\) , \( -9528 a - 620\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(764a-756\right){x}-9528a-620$ |
| 42592.18-f7 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{23} \cdot 11^{7} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.760210303$ |
2.298659893 |
\( -\frac{7153263}{2816} a + \frac{88973461}{2816} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -24 a + 161\) , \( 487 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-24a+161\right){x}+487a+3$ |
| 42592.18-f8 |
42592.18-f |
$8$ |
$12$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{17} \cdot 11^{10} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.380105151$ |
2.298659893 |
\( \frac{67333244623}{117128} a + \frac{61299754338}{14641} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -1011 a + 449\) , \( -10902 a + 18942\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-1011a+449\right){x}-10902a+18942$ |
| 42592.18-g1 |
42592.18-g |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{23} \cdot 11^{7} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 3^{2} \cdot 13 \) |
$0.007380428$ |
$0.900422072$ |
9.404041244 |
\( -\frac{3287573457}{10903552} a + \frac{14322326347}{10903552} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -30 a - 22\) , \( -9 a - 81\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-30a-22\right){x}-9a-81$ |
| 42592.18-h1 |
42592.18-h |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{11} \cdot 11^{5} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.129524659$ |
$2.156487348$ |
7.601212072 |
\( \frac{3132447}{88} a - \frac{4697477}{88} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -16 a + 17\) , \( a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-16a+17\right){x}+a+31$ |
| 42592.18-i1 |
42592.18-i |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{13} \cdot 11^{11} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.278106551$ |
$0.658252719$ |
8.303020439 |
\( \frac{35786999}{42592} a + \frac{381275699}{42592} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 70 a - 166\) , \( 424 a - 620\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(70a-166\right){x}+424a-620$ |
| 42592.18-j1 |
42592.18-j |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{5} \cdot 11^{11} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$0.688611849$ |
$0.960810471$ |
9.002553276 |
\( \frac{51319633286103}{4715895382} a - \frac{160385954000237}{4715895382} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -69 a + 58\) , \( -102 a + 470\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-69a+58\right){x}-102a+470$ |
| 42592.18-j2 |
42592.18-j |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{7} \cdot 11^{5} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$0.229537283$ |
$2.882431414$ |
9.002553276 |
\( -\frac{78558129}{10648} a - \frac{212320277}{10648} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( a + 8\) , \( -4 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(a+8\right){x}-4a+4$ |
| 42592.18-k1 |
42592.18-k |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{21} \cdot 11^{9} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 3 \cdot 7 \) |
$0.104477125$ |
$0.649149721$ |
8.613037707 |
\( \frac{399175}{1408} a + \frac{5454499}{1408} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 30 a + 115\) , \( -223 a + 363\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(30a+115\right){x}-223a+363$ |
| 42592.18-l1 |
42592.18-l |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
42592.18 |
\( 2^{5} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{9} \) |
$3.39641$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 5 \) |
$1$ |
$1.349519396$ |
5.100703873 |
\( -\frac{82377}{352} a + \frac{639603}{352} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 18 a - 3\) , \( 13 a - 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(18a-3\right){x}+13a-33$ |
*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.
