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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 425880o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
425880.o4 | 425880o1 | \([0, 0, 0, 13182, 4809233]\) | \(4499456/180075\) | \(-10138204524193200\) | \([2]\) | \(2359296\) | \(1.7511\) | \(\Gamma_0(N)\)-optimal* |
425880.o3 | 425880o2 | \([0, 0, 0, -359463, 79412762]\) | \(5702413264/275625\) | \(248282559776160000\) | \([2, 2]\) | \(4718592\) | \(2.0977\) | \(\Gamma_0(N)\)-optimal* |
425880.o1 | 425880o3 | \([0, 0, 0, -5682963, 5214460862]\) | \(5633270409316/14175\) | \(51075269439667200\) | \([2]\) | \(9437184\) | \(2.4443\) | \(\Gamma_0(N)\)-optimal* |
425880.o2 | 425880o4 | \([0, 0, 0, -998283, -281009482]\) | \(30534944836/8203125\) | \(29557447592400000000\) | \([2]\) | \(9437184\) | \(2.4443\) |
Rank
sage: E.rank()
The elliptic curves in class 425880o have rank \(2\).
Complex multiplication
The elliptic curves in class 425880o do not have complex multiplication.Modular form 425880.2.a.o
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.