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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 14400bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14400.fd3 | 14400bp1 | \([0, 0, 0, -1800, -27000]\) | \(55296/5\) | \(58320000000\) | \([2]\) | \(12288\) | \(0.80524\) | \(\Gamma_0(N)\)-optimal |
14400.fd2 | 14400bp2 | \([0, 0, 0, -6300, 162000]\) | \(148176/25\) | \(4665600000000\) | \([2, 2]\) | \(24576\) | \(1.1518\) | |
14400.fd1 | 14400bp3 | \([0, 0, 0, -96300, 11502000]\) | \(132304644/5\) | \(3732480000000\) | \([2]\) | \(49152\) | \(1.4984\) | |
14400.fd4 | 14400bp4 | \([0, 0, 0, 11700, 918000]\) | \(237276/625\) | \(-466560000000000\) | \([2]\) | \(49152\) | \(1.4984\) |
Rank
sage: E.rank()
The elliptic curves in class 14400bp have rank \(0\).
Complex multiplication
The elliptic curves in class 14400bp do not have complex multiplication.Modular form 14400.2.a.bp
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.