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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 190575ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
190575.dh4 | 190575ed1 | \([1, -1, 0, -7215192, 7461440091]\) | \(2058561081361/12705\) | \(256376571033515625\) | \([2]\) | \(5898240\) | \(2.5271\) | \(\Gamma_0(N)\)-optimal |
190575.dh3 | 190575ed2 | \([1, -1, 0, -7351317, 7165368216]\) | \(2177286259681/161417025\) | \(3257264334980816015625\) | \([2, 2]\) | \(11796480\) | \(2.8737\) | |
190575.dh5 | 190575ed3 | \([1, -1, 0, 6941808, 31649491341]\) | \(1833318007919/22507682505\) | \(-454186734556705974140625\) | \([2]\) | \(23592960\) | \(3.2203\) | |
190575.dh2 | 190575ed4 | \([1, -1, 0, -23822442, -36268988409]\) | \(74093292126001/14707625625\) | \(296787928042673525390625\) | \([2, 2]\) | \(23592960\) | \(3.2203\) | |
190575.dh6 | 190575ed5 | \([1, -1, 0, 49548933, -215662000284]\) | \(666688497209279/1381398046875\) | \(-27875489530909735107421875\) | \([2]\) | \(47185920\) | \(3.5668\) | |
190575.dh1 | 190575ed6 | \([1, -1, 0, -360731817, -2636872454034]\) | \(257260669489908001/14267882475\) | \(287914268779983488671875\) | \([2]\) | \(47185920\) | \(3.5668\) |
Rank
sage: E.rank()
The elliptic curves in class 190575ed have rank \(1\).
Complex multiplication
The elliptic curves in class 190575ed do not have complex multiplication.Modular form 190575.2.a.ed
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.