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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 78400ij
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78400.bu2 | 78400ij1 | \([0, 1, 0, -513, 3583]\) | \(46585/8\) | \(2569011200\) | \([]\) | \(41472\) | \(0.52594\) | \(\Gamma_0(N)\)-optimal |
78400.bu1 | 78400ij2 | \([0, 1, 0, -11713, -491457]\) | \(553463785/512\) | \(164416716800\) | \([]\) | \(124416\) | \(1.0752\) |
Rank
sage: E.rank()
The elliptic curves in class 78400ij have rank \(1\).
Complex multiplication
The elliptic curves in class 78400ij do not have complex multiplication.Modular form 78400.2.a.ij
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.