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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 12675bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12675.e1 | 12675bn1 | \([0, 1, 1, -628, 5854]\) | \(-99897344/27\) | \(-7414875\) | \([]\) | \(7488\) | \(0.30150\) | \(\Gamma_0(N)\)-optimal |
12675.e2 | 12675bn2 | \([0, 1, 1, 3922, -12346]\) | \(24288219136/14348907\) | \(-3940568584875\) | \([]\) | \(37440\) | \(1.1062\) |
Rank
sage: E.rank()
The elliptic curves in class 12675bn have rank \(2\).
Complex multiplication
The elliptic curves in class 12675bn do not have complex multiplication.Modular form 12675.2.a.bn
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.