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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 490d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
490.b2 | 490d1 | \([1, 1, 0, 3, 1]\) | \(34391/20\) | \(-980\) | \([]\) | \(24\) | \(-0.73660\) | \(\Gamma_0(N)\)-optimal |
490.b1 | 490d2 | \([1, 1, 0, -32, 64]\) | \(-77626969/8000\) | \(-392000\) | \([]\) | \(72\) | \(-0.18729\) |
Rank
sage: E.rank()
The elliptic curves in class 490d have rank \(1\).
Complex multiplication
The elliptic curves in class 490d do not have complex multiplication.Modular form 490.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.