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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1008d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1008.f2 | 1008d1 | \([0, 0, 0, -6, -9]\) | \(-55296/49\) | \(-21168\) | \([2]\) | \(64\) | \(-0.47287\) | \(\Gamma_0(N)\)-optimal |
1008.f1 | 1008d2 | \([0, 0, 0, -111, -450]\) | \(21882096/7\) | \(48384\) | \([2]\) | \(128\) | \(-0.12630\) |
Rank
sage: E.rank()
The elliptic curves in class 1008d have rank \(0\).
Complex multiplication
The elliptic curves in class 1008d do not have complex multiplication.Modular form 1008.2.a.d
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.