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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1584.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1584.d1 | 1584g3 | \([0, 0, 0, -4251, 106666]\) | \(5690357426/891\) | \(1330255872\) | \([2]\) | \(1024\) | \(0.76172\) | |
1584.d2 | 1584g2 | \([0, 0, 0, -291, 1330]\) | \(3650692/1089\) | \(812934144\) | \([2, 2]\) | \(512\) | \(0.41515\) | |
1584.d3 | 1584g1 | \([0, 0, 0, -111, -434]\) | \(810448/33\) | \(6158592\) | \([2]\) | \(256\) | \(0.068572\) | \(\Gamma_0(N)\)-optimal |
1584.d4 | 1584g4 | \([0, 0, 0, 789, 8890]\) | \(36382894/43923\) | \(-65576687616\) | \([4]\) | \(1024\) | \(0.76172\) |
Rank
sage: E.rank()
The elliptic curves in class 1584.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1584.d do not have complex multiplication.Modular form 1584.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.