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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 82810d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82810.t2 | 82810d1 | \([1, -1, 0, 40, -64]\) | \(842751/640\) | \(-5299840\) | \([]\) | \(15120\) | \(-0.021602\) | \(\Gamma_0(N)\)-optimal |
82810.t1 | 82810d2 | \([1, -1, 0, -13610, 614550]\) | \(-33669235266849/156250\) | \(-1293906250\) | \([]\) | \(105840\) | \(0.95135\) |
Rank
sage: E.rank()
The elliptic curves in class 82810d have rank \(0\).
Complex multiplication
The elliptic curves in class 82810d do not have complex multiplication.Modular form 82810.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.