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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 333270.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.bj1 | 333270bj4 | \([1, -1, 0, -2296516059, -41455596567987]\) | \(12411881707829361287041/303132494474220600\) | \(32713501973819270061169668600\) | \([2]\) | \(525533184\) | \(4.2555\) | |
333270.bj2 | 333270bj2 | \([1, -1, 0, -282613059, 1806613478613]\) | \(23131609187144855041/322060536000000\) | \(34756181446002271416000000\) | \([2]\) | \(175177728\) | \(3.7062\) | |
333270.bj3 | 333270bj1 | \([1, -1, 0, -2285379, 75702185685]\) | \(-12232183057921/22933241856000\) | \(-2474913334591823118336000\) | \([2]\) | \(87588864\) | \(3.3597\) | \(\Gamma_0(N)\)-optimal |
333270.bj4 | 333270bj3 | \([1, -1, 0, 20567421, -2043396823275]\) | \(8915971454369279/16719623332762560\) | \(-1804351037477962680331047360\) | \([2]\) | \(262766592\) | \(3.9090\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.bj have rank \(2\).
Complex multiplication
The elliptic curves in class 333270.bj do not have complex multiplication.Modular form 333270.2.a.bj
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.