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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 223440bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223440.ec4 | 223440bz1 | \([0, 1, 0, 7318624, -16201668876]\) | \(89962967236397039/287450726400000\) | \(-138519717929916825600000\) | \([2]\) | \(20736000\) | \(3.1236\) | \(\Gamma_0(N)\)-optimal |
223440.ec3 | 223440bz2 | \([0, 1, 0, -68948896, -189878065420]\) | \(75224183150104868881/11219310000000000\) | \(5406476706570240000000000\) | \([2]\) | \(41472000\) | \(3.4702\) | |
223440.ec2 | 223440bz3 | \([0, 1, 0, -2588348576, -50686206591756]\) | \(-3979640234041473454886161/1471455901872240\) | \(-709080331875807902760960\) | \([2]\) | \(103680000\) | \(3.9284\) | |
223440.ec1 | 223440bz4 | \([0, 1, 0, -41413580896, -3243875203796620]\) | \(16300610738133468173382620881/2228489100\) | \(1073887289859686400\) | \([2]\) | \(207360000\) | \(4.2749\) |
Rank
sage: E.rank()
The elliptic curves in class 223440bz have rank \(0\).
Complex multiplication
The elliptic curves in class 223440bz do not have complex multiplication.Modular form 223440.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.