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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 27200b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27200.bo1 | 27200b1 | \([0, 0, 0, -2800, 57000]\) | \(151732224/85\) | \(1360000000\) | \([2]\) | \(18432\) | \(0.69975\) | \(\Gamma_0(N)\)-optimal |
27200.bo2 | 27200b2 | \([0, 0, 0, -2300, 78000]\) | \(-5256144/7225\) | \(-1849600000000\) | \([2]\) | \(36864\) | \(1.0463\) |
Rank
sage: E.rank()
The elliptic curves in class 27200b have rank \(1\).
Complex multiplication
The elliptic curves in class 27200b do not have complex multiplication.Modular form 27200.2.a.b
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.