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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 259920.cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.cv1 | 259920cv4 | \([0, 0, 0, -2745973965963, 1751429807915291962]\) | \(16300610738133468173382620881/2228489100\) | \(313054244303345606246400\) | \([2]\) | \(1658880000\) | \(5.3235\) | |
259920.cv2 | 259920cv3 | \([0, 0, 0, -171623357643, 27366095848399738]\) | \(-3979640234041473454886161/1471455901872240\) | \(-206707547004071933941627944960\) | \([2]\) | \(829440000\) | \(4.9769\) | |
259920.cv3 | 259920cv2 | \([0, 0, 0, -4571733963, 102506799664762]\) | \(75224183150104868881/11219310000000000\) | \(1576069011804889866240000000000\) | \([2]\) | \(331776000\) | \(4.5188\) | |
259920.cv4 | 259920cv1 | \([0, 0, 0, 485269557, 8748943003258]\) | \(89962967236397039/287450726400000\) | \(-40380574411425102535065600000\) | \([2]\) | \(165888000\) | \(4.1722\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259920.cv have rank \(0\).
Complex multiplication
The elliptic curves in class 259920.cv do not have complex multiplication.Modular form 259920.2.a.cv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.