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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 585.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
585.f1 | 585d1 | \([0, 0, 1, -42, 105]\) | \(-303464448/1625\) | \(-43875\) | \([3]\) | \(48\) | \(-0.26471\) | \(\Gamma_0(N)\)-optimal |
585.f2 | 585d2 | \([0, 0, 1, 108, 560]\) | \(7077888/10985\) | \(-216217755\) | \([]\) | \(144\) | \(0.28459\) |
Rank
sage: E.rank()
The elliptic curves in class 585.f have rank \(1\).
Complex multiplication
The elliptic curves in class 585.f do not have complex multiplication.Modular form 585.2.a.f
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.