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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 31680dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.cl2 | 31680dm1 | \([0, 0, 0, 5748, -677104]\) | \(109902239/1100000\) | \(-210213273600000\) | \([]\) | \(115200\) | \(1.4294\) | \(\Gamma_0(N)\)-optimal |
31680.cl1 | 31680dm2 | \([0, 0, 0, -3421452, -2435924464]\) | \(-23178622194826561/1610510\) | \(-307773253877760\) | \([]\) | \(576000\) | \(2.2341\) |
Rank
sage: E.rank()
The elliptic curves in class 31680dm have rank \(0\).
Complex multiplication
The elliptic curves in class 31680dm do not have complex multiplication.Modular form 31680.2.a.dm
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.