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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5082j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5082.l4 | 5082j1 | \([1, 0, 1, 9314, -204976]\) | \(50447927375/39517632\) | \(-70007895663552\) | \([2]\) | \(11520\) | \(1.3446\) | \(\Gamma_0(N)\)-optimal |
| 5082.l3 | 5082j2 | \([1, 0, 1, -43926, -1780880]\) | \(5290763640625/2291573592\) | \(4059662404217112\) | \([2]\) | \(23040\) | \(1.6912\) | |
| 5082.l2 | 5082j3 | \([1, 0, 1, -99586, 15354656]\) | \(-61653281712625/21875235228\) | \(-38753313595750908\) | \([2]\) | \(34560\) | \(1.8939\) | |
| 5082.l1 | 5082j4 | \([1, 0, 1, -1710096, 860550304]\) | \(312196988566716625/25367712678\) | \(44940450439550358\) | \([2]\) | \(69120\) | \(2.2405\) |
Rank
sage: E.rank()
The elliptic curves in class 5082j have rank \(1\).
Complex multiplication
The elliptic curves in class 5082j do not have complex multiplication.Modular form 5082.2.a.j
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
