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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2034.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2034.d1 | 2034c2 | \([1, -1, 0, -64089, 69319881]\) | \(-39934705050538129/2823126576537804\) | \(-2058059274296059116\) | \([]\) | \(28224\) | \(2.1938\) | |
2034.d2 | 2034c1 | \([1, -1, 0, -14949, -704619]\) | \(-506814405937489/4048994304\) | \(-2951716847616\) | \([]\) | \(4032\) | \(1.2208\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2034.d have rank \(0\).
Complex multiplication
The elliptic curves in class 2034.d do not have complex multiplication.Modular form 2034.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.