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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 40425.dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40425.dd1 | 40425cs2 | \([0, 1, 1, -32818158, 1368065726969]\) | \(-2126464142970105856/438611057788643355\) | \(-806283630277751594881171875\) | \([]\) | \(34560000\) | \(3.8422\) | |
40425.dd2 | 40425cs1 | \([0, 1, 1, -10951908, -16340583031]\) | \(-79028701534867456/16987307596875\) | \(-31227183616636669921875\) | \([]\) | \(6912000\) | \(3.0374\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40425.dd have rank \(0\).
Complex multiplication
The elliptic curves in class 40425.dd do not have complex multiplication.Modular form 40425.2.a.dd
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.