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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 32490bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32490.bl4 | 32490bp1 | \([1, -1, 1, -32558, 3866541]\) | \(-111284641/123120\) | \(-4222578585296880\) | \([2]\) | \(276480\) | \(1.6929\) | \(\Gamma_0(N)\)-optimal |
32490.bl3 | 32490bp2 | \([1, -1, 1, -617378, 186798237]\) | \(758800078561/324900\) | \(11142915711200100\) | \([2, 2]\) | \(552960\) | \(2.0395\) | |
32490.bl2 | 32490bp3 | \([1, -1, 1, -714848, 123949581]\) | \(1177918188481/488703750\) | \(16760802382263483750\) | \([2]\) | \(1105920\) | \(2.3860\) | |
32490.bl1 | 32490bp4 | \([1, -1, 1, -9877028, 11950257597]\) | \(3107086841064961/570\) | \(19548974931930\) | \([2]\) | \(1105920\) | \(2.3860\) |
Rank
sage: E.rank()
The elliptic curves in class 32490bp have rank \(0\).
Complex multiplication
The elliptic curves in class 32490bp do not have complex multiplication.Modular form 32490.2.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.