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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 18240cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18240.bo4 | 18240cm1 | \([0, 1, 0, -641, -10881]\) | \(-111284641/123120\) | \(-32275169280\) | \([2]\) | \(18432\) | \(0.71109\) | \(\Gamma_0(N)\)-optimal |
18240.bo3 | 18240cm2 | \([0, 1, 0, -12161, -520065]\) | \(758800078561/324900\) | \(85170585600\) | \([2, 2]\) | \(36864\) | \(1.0577\) | |
18240.bo1 | 18240cm3 | \([0, 1, 0, -194561, -33096705]\) | \(3107086841064961/570\) | \(149422080\) | \([2]\) | \(73728\) | \(1.4042\) | |
18240.bo2 | 18240cm4 | \([0, 1, 0, -14081, -346881]\) | \(1177918188481/488703750\) | \(128110755840000\) | \([2]\) | \(73728\) | \(1.4042\) |
Rank
sage: E.rank()
The elliptic curves in class 18240cm have rank \(1\).
Complex multiplication
The elliptic curves in class 18240cm do not have complex multiplication.Modular form 18240.2.a.cm
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.