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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 34200.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34200.bp1 | 34200z4 | \([0, 0, 0, -6840075, -6885562250]\) | \(3034301922374404/1425\) | \(16621200000000\) | \([2]\) | \(393216\) | \(2.3105\) | |
34200.bp2 | 34200z3 | \([0, 0, 0, -513075, -61465250]\) | \(1280615525284/601171875\) | \(7012068750000000000\) | \([2]\) | \(393216\) | \(2.3105\) | |
34200.bp3 | 34200z2 | \([0, 0, 0, -427575, -107549750]\) | \(2964647793616/2030625\) | \(5921302500000000\) | \([2, 2]\) | \(196608\) | \(1.9639\) | |
34200.bp4 | 34200z1 | \([0, 0, 0, -21450, -2363375]\) | \(-5988775936/9774075\) | \(-1781325168750000\) | \([2]\) | \(98304\) | \(1.6173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34200.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 34200.bp do not have complex multiplication.Modular form 34200.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.