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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 69696.di
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.di1 | 69696bl1 | \([0, 0, 0, -36300, 787952]\) | \(62500/33\) | \(2793042278940672\) | \([2]\) | \(245760\) | \(1.6554\) | \(\Gamma_0(N)\)-optimal |
69696.di2 | 69696bl2 | \([0, 0, 0, 137940, 6154544]\) | \(1714750/1089\) | \(-184340790410084352\) | \([2]\) | \(491520\) | \(2.0020\) |
Rank
sage: E.rank()
The elliptic curves in class 69696.di have rank \(1\).
Complex multiplication
The elliptic curves in class 69696.di do not have complex multiplication.Modular form 69696.2.a.di
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.