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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 230115b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
230115.b4 | 230115b1 | \([1, 1, 1, -2777261, 1732668458]\) | \(16003198512756001/488525390625\) | \(72319290500244140625\) | \([2]\) | \(6488064\) | \(2.5866\) | \(\Gamma_0(N)\)-optimal |
230115.b2 | 230115b2 | \([1, 1, 1, -44105386, 112723480958]\) | \(64096096056024006001/62562515625\) | \(9261497618623265625\) | \([2, 2]\) | \(12976128\) | \(2.9332\) | |
230115.b1 | 230115b3 | \([1, 1, 1, -705686011, 7215188438708]\) | \(262537424941059264096001/250125\) | \(37027476736125\) | \([2]\) | \(25952256\) | \(3.2797\) | |
230115.b3 | 230115b4 | \([1, 1, 1, -43774761, 114497085708]\) | \(-62665433378363916001/2004003001000125\) | \(-296664365811721393486125\) | \([2]\) | \(25952256\) | \(3.2797\) |
Rank
sage: E.rank()
The elliptic curves in class 230115b have rank \(1\).
Complex multiplication
The elliptic curves in class 230115b do not have complex multiplication.Modular form 230115.2.a.b
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.