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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 30030.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30030.p1 | 30030r8 | \([1, 0, 1, -105752574, -418583571128]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(4007004103295286093000\) | \([2]\) | \(5971968\) | \(3.2426\) | |
| 30030.p2 | 30030r6 | \([1, 0, 1, -6887574, -5960607128]\) | \(36134533748915083453404889/5565686539253841000000\) | \(5565686539253841000000\) | \([2, 2]\) | \(2985984\) | \(2.8961\) | |
| 30030.p3 | 30030r5 | \([1, 0, 1, -2313909, 426827206]\) | \(1370131553911340548947529/714126686285699857170\) | \(714126686285699857170\) | \([6]\) | \(1990656\) | \(2.6933\) | |
| 30030.p4 | 30030r3 | \([1, 0, 1, -1887574, 907392872]\) | \(743764321292317933404889/74603529000000000000\) | \(74603529000000000000\) | \([2]\) | \(1492992\) | \(2.5495\) | |
| 30030.p5 | 30030r2 | \([1, 0, 1, -1840059, 959624146]\) | \(688999042618248810121129/779639711718968100\) | \(779639711718968100\) | \([2, 6]\) | \(995328\) | \(2.3467\) | |
| 30030.p6 | 30030r1 | \([1, 0, 1, -1839559, 960172346]\) | \(688437529087783927489129/882972090000\) | \(882972090000\) | \([6]\) | \(497664\) | \(2.0002\) | \(\Gamma_0(N)\)-optimal |
| 30030.p7 | 30030r4 | \([1, 0, 1, -1374209, 1457338286]\) | \(-286999819333751016766729/751553009101890965970\) | \(-751553009101890965970\) | \([6]\) | \(1990656\) | \(2.6933\) | |
| 30030.p8 | 30030r7 | \([1, 0, 1, 11977426, -32869643128]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-576736012771479654093000\) | \([2]\) | \(5971968\) | \(3.2426\) |
Rank
sage: E.rank()
The elliptic curves in class 30030.p have rank \(1\).
Complex multiplication
The elliptic curves in class 30030.p do not have complex multiplication.Modular form 30030.2.a.p
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
