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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 485100.eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.eh1 | 485100eh2 | \([0, 0, 0, -287175, -80239250]\) | \(-18330740176/8857805\) | \(-1265638609620000000\) | \([]\) | \(7464960\) | \(2.1791\) | |
485100.eh2 | 485100eh1 | \([0, 0, 0, 27825, 1345750]\) | \(16674224/15125\) | \(-2161120500000000\) | \([]\) | \(2488320\) | \(1.6298\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 485100.eh do not have complex multiplication.Modular form 485100.2.a.eh
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.