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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 138600dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
138600.ee3 | 138600dm1 | \([0, 0, 0, -16050, -667375]\) | \(2508888064/396165\) | \(72201071250000\) | \([2]\) | \(294912\) | \(1.3820\) | \(\Gamma_0(N)\)-optimal |
138600.ee2 | 138600dm2 | \([0, 0, 0, -71175, 6664250]\) | \(13674725584/1334025\) | \(3890016900000000\) | \([2, 2]\) | \(589824\) | \(1.7285\) | |
138600.ee1 | 138600dm3 | \([0, 0, 0, -1110675, 450530750]\) | \(12990838708516/144375\) | \(1683990000000000\) | \([2]\) | \(1179648\) | \(2.0751\) | |
138600.ee4 | 138600dm4 | \([0, 0, 0, 86325, 32021750]\) | \(6099383804/41507235\) | \(-484140389040000000\) | \([2]\) | \(1179648\) | \(2.0751\) |
Rank
sage: E.rank()
The elliptic curves in class 138600dm have rank \(0\).
Complex multiplication
The elliptic curves in class 138600dm do not have complex multiplication.Modular form 138600.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.