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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 6240.v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6240.v1 | 6240l2 | \([0, 1, 0, -25625841, -49938972705]\) | \(454357982636417669333824/3003024375\) | \(12300387840000\) | \([2]\) | \(215040\) | \(2.5707\) | |
6240.v2 | 6240l3 | \([0, 1, 0, -1711496, -667563336]\) | \(1082883335268084577352/251301565117746585\) | \(128666401340286251520\) | \([2]\) | \(215040\) | \(2.5707\) | |
6240.v3 | 6240l1 | \([0, 1, 0, -1601646, -780664896]\) | \(7099759044484031233216/577161945398025\) | \(36938364505473600\) | \([2, 2]\) | \(107520\) | \(2.2241\) | \(\Gamma_0(N)\)-optimal |
6240.v4 | 6240l4 | \([0, 1, 0, -1492296, -891720756]\) | \(-717825640026599866952/254764560814329735\) | \(-130439455136936824320\) | \([2]\) | \(215040\) | \(2.5707\) |
Rank
sage: E.rank()
The elliptic curves in class 6240.v have rank \(1\).
Complex multiplication
The elliptic curves in class 6240.v do not have complex multiplication.Modular form 6240.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.