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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 69696eu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.gu2 | 69696eu1 | \([0, 0, 0, -13068, -106480]\) | \(19683/11\) | \(137928013774848\) | \([2]\) | \(245760\) | \(1.4034\) | \(\Gamma_0(N)\)-optimal |
69696.gu1 | 69696eu2 | \([0, 0, 0, -129228, 17782160]\) | \(19034163/121\) | \(1517208151523328\) | \([2]\) | \(491520\) | \(1.7500\) |
Rank
sage: E.rank()
The elliptic curves in class 69696eu have rank \(1\).
Complex multiplication
The elliptic curves in class 69696eu do not have complex multiplication.Modular form 69696.2.a.eu
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.