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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 2800bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2800.s1 | 2800bc1 | \([0, 0, 0, -2000, 34375]\) | \(28311552/49\) | \(1531250000\) | \([2]\) | \(1440\) | \(0.65639\) | \(\Gamma_0(N)\)-optimal |
2800.s2 | 2800bc2 | \([0, 0, 0, -1375, 56250]\) | \(-574992/2401\) | \(-1200500000000\) | \([2]\) | \(2880\) | \(1.0030\) |
Rank
sage: E.rank()
The elliptic curves in class 2800bc have rank \(0\).
Complex multiplication
The elliptic curves in class 2800bc do not have complex multiplication.Modular form 2800.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.