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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 585.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
585.d1 | 585a2 | \([1, -1, 1, -3593, 83782]\) | \(260549802603/4225\) | \(83160675\) | \([2]\) | \(384\) | \(0.65208\) | |
585.d2 | 585a1 | \([1, -1, 1, -218, 1432]\) | \(-57960603/8125\) | \(-159924375\) | \([2]\) | \(192\) | \(0.30551\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 585.d have rank \(1\).
Complex multiplication
The elliptic curves in class 585.d do not have complex multiplication.Modular form 585.2.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 LMFDB 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 LMFDB labels.