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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 10710d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10710.f2 | 10710d1 | \([1, -1, 0, 21, 1285]\) | \(36926037/26656000\) | \(-719712000\) | \([2]\) | \(4608\) | \(0.37857\) | \(\Gamma_0(N)\)-optimal |
10710.f1 | 10710d2 | \([1, -1, 0, -1659, 25813]\) | \(18708817969323/505750000\) | \(13655250000\) | \([2]\) | \(9216\) | \(0.72514\) |
Rank
sage: E.rank()
The elliptic curves in class 10710d have rank \(1\).
Complex multiplication
The elliptic curves in class 10710d do not have complex multiplication.Modular form 10710.2.a.d
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.