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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 45177.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45177.d1 | 45177n2 | \([1, 0, 0, -142830, 20761191]\) | \(6361756446348757/1291467969\) | \(65416727033757\) | \([2]\) | \(352512\) | \(1.6487\) | |
45177.d2 | 45177n1 | \([1, 0, 0, -7965, 396576]\) | \(-1103263596037/707347971\) | \(-35829296775063\) | \([2]\) | \(176256\) | \(1.3021\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 45177.d have rank \(2\).
Complex multiplication
The elliptic curves in class 45177.d do not have complex multiplication.Modular form 45177.2.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 LMFDB 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 LMFDB labels.