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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 414050.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.df1 | 414050df2 | \([1, 1, 0, -1515595, -718220355]\) | \(553463785/512\) | \(356167594880115200\) | \([]\) | \(10450944\) | \(2.2910\) | |
414050.df2 | 414050df1 | \([1, 1, 0, -66420, 5497640]\) | \(46585/8\) | \(5565118670001800\) | \([]\) | \(3483648\) | \(1.7416\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050.df have rank \(1\).
Complex multiplication
The elliptic curves in class 414050.df do not have complex multiplication.Modular form 414050.2.a.df
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.