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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 1710m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.p2 | 1710m1 | \([1, -1, 1, -62, -51]\) | \(961504803/486400\) | \(13132800\) | \([2]\) | \(640\) | \(0.058433\) | \(\Gamma_0(N)\)-optimal |
1710.p1 | 1710m2 | \([1, -1, 1, -542, 4941]\) | \(651038076963/7220000\) | \(194940000\) | \([2]\) | \(1280\) | \(0.40501\) |
Rank
sage: E.rank()
The elliptic curves in class 1710m have rank \(1\).
Complex multiplication
The elliptic curves in class 1710m do not have complex multiplication.Modular form 1710.2.a.m
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.