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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 56784v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56784.ce4 | 56784v1 | \([0, 1, 0, -1239, -109368]\) | \(-2725888/64827\) | \(-5006520752688\) | \([2]\) | \(92160\) | \(1.1175\) | \(\Gamma_0(N)\)-optimal |
56784.ce3 | 56784v2 | \([0, 1, 0, -42644, -3388644]\) | \(6940769488/35721\) | \(44139121737984\) | \([2, 2]\) | \(184320\) | \(1.4641\) | |
56784.ce2 | 56784v3 | \([0, 1, 0, -66304, 766052]\) | \(6522128932/3720087\) | \(18387096998280192\) | \([2]\) | \(368640\) | \(1.8107\) | |
56784.ce1 | 56784v4 | \([0, 1, 0, -681464, -216754524]\) | \(7080974546692/189\) | \(934161306624\) | \([2]\) | \(368640\) | \(1.8107\) |
Rank
sage: E.rank()
The elliptic curves in class 56784v have rank \(1\).
Complex multiplication
The elliptic curves in class 56784v do not have complex multiplication.Modular form 56784.2.a.v
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.