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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 235200.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.r1 | 235200r3 | \([0, -1, 0, -1476533, 638332437]\) | \(189123395584/16078125\) | \(30265205250000000000\) | \([2]\) | \(7962624\) | \(2.4795\) | |
235200.r2 | 235200r1 | \([0, -1, 0, -300533, -63151563]\) | \(1594753024/4725\) | \(8894264400000000\) | \([2]\) | \(2654208\) | \(1.9301\) | \(\Gamma_0(N)\)-optimal |
235200.r3 | 235200r2 | \([0, -1, 0, -178033, -115214063]\) | \(-20720464/178605\) | \(-5379251109120000000\) | \([2]\) | \(5308416\) | \(2.2767\) | |
235200.r4 | 235200r4 | \([0, -1, 0, 1585967, 2938269937]\) | \(14647977776/132355125\) | \(-3986290713888000000000\) | \([2]\) | \(15925248\) | \(2.8260\) |
Rank
sage: E.rank()
The elliptic curves in class 235200.r have rank \(0\).
Complex multiplication
The elliptic curves in class 235200.r do not have complex multiplication.Modular form 235200.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.