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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 52800cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52800.fk3 | 52800cr1 | \([0, 1, 0, -3308, 72138]\) | \(4004529472/99\) | \(99000000\) | \([2]\) | \(32768\) | \(0.64337\) | \(\Gamma_0(N)\)-optimal |
| 52800.fk2 | 52800cr2 | \([0, 1, 0, -3433, 66263]\) | \(69934528/9801\) | \(627264000000\) | \([2, 2]\) | \(65536\) | \(0.98994\) | |
| 52800.fk4 | 52800cr3 | \([0, 1, 0, 5567, 363263]\) | \(37259704/131769\) | \(-67465728000000\) | \([2]\) | \(131072\) | \(1.3365\) | |
| 52800.fk1 | 52800cr4 | \([0, 1, 0, -14433, -604737]\) | \(649461896/72171\) | \(36951552000000\) | \([2]\) | \(131072\) | \(1.3365\) |
Rank
sage: E.rank()
The elliptic curves in class 52800cr have rank \(1\).
Complex multiplication
The elliptic curves in class 52800cr do not have complex multiplication.Modular form 52800.2.a.cr
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.
