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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 69696.he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.he1 | 69696do1 | \([0, 0, 0, -34919148, -79422473680]\) | \(55635379958596/24057\) | \(2036127821347749888\) | \([2]\) | \(5160960\) | \(2.8555\) | \(\Gamma_0(N)\)-optimal |
| 69696.he2 | 69696do2 | \([0, 0, 0, -34744908, -80254295440]\) | \(-27403349188178/578739249\) | \(-97966253996325638111232\) | \([2]\) | \(10321920\) | \(3.2021\) |
Rank
sage: E.rank()
The elliptic curves in class 69696.he have rank \(1\).
Complex multiplication
The elliptic curves in class 69696.he do not have complex multiplication.Modular form 69696.2.a.he
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
