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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 372810.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 372810.f1 | 372810f3 | \([1, 1, 0, -106059393, 420364601397]\) | \(5466100368607268326201/465445828800\) | \(11234730808422187200\) | \([2]\) | \(46006272\) | \(3.0970\) | |
| 372810.f2 | 372810f2 | \([1, 1, 0, -6643393, 6535559797]\) | \(1343383839781990201/12311677440000\) | \(297173963713743360000\) | \([2, 2]\) | \(23003136\) | \(2.7504\) | |
| 372810.f3 | 372810f4 | \([1, 1, 0, -1926913, 15639309493]\) | \(-32780596813828921/4358971275000000\) | \(-105214969919330475000000\) | \([2]\) | \(46006272\) | \(3.0970\) | |
| 372810.f4 | 372810f1 | \([1, 1, 0, -724673, -70915467]\) | \(1743642162605881/919810867200\) | \(22201998273989836800\) | \([2]\) | \(11501568\) | \(2.4038\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 372810.f have rank \(1\).
Complex multiplication
The elliptic curves in class 372810.f do not have complex multiplication.Modular form 372810.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
