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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 27225.bx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27225.bx1 | 27225bq3 | \([0, 0, 1, -212908575, -1195742989719]\) | \(-52893159101157376/11\) | \(-221971057171875\) | \([]\) | \(2520000\) | \(3.0497\) | |
27225.bx2 | 27225bq2 | \([0, 0, 1, -281325, -104025969]\) | \(-122023936/161051\) | \(-3249878248053421875\) | \([]\) | \(504000\) | \(2.2450\) | |
27225.bx3 | 27225bq1 | \([0, 0, 1, -9075, 790281]\) | \(-4096/11\) | \(-221971057171875\) | \([]\) | \(100800\) | \(1.4402\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27225.bx have rank \(1\).
Complex multiplication
The elliptic curves in class 27225.bx do not have complex multiplication.Modular form 27225.2.a.bx
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.