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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 4200o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.bb4 | 4200o1 | \([0, 1, 0, -4383, 110238]\) | \(37256083456/525\) | \(131250000\) | \([2]\) | \(3072\) | \(0.69791\) | \(\Gamma_0(N)\)-optimal |
| 4200.bb3 | 4200o2 | \([0, 1, 0, -4508, 103488]\) | \(2533446736/275625\) | \(1102500000000\) | \([2, 2]\) | \(6144\) | \(1.0445\) | |
| 4200.bb2 | 4200o3 | \([0, 1, 0, -17008, -746512]\) | \(34008619684/4862025\) | \(77792400000000\) | \([2, 2]\) | \(12288\) | \(1.3911\) | |
| 4200.bb5 | 4200o4 | \([0, 1, 0, 5992, 523488]\) | \(1486779836/8203125\) | \(-131250000000000\) | \([2]\) | \(12288\) | \(1.3911\) | |
| 4200.bb1 | 4200o5 | \([0, 1, 0, -262008, -51706512]\) | \(62161150998242/1607445\) | \(51438240000000\) | \([2]\) | \(24576\) | \(1.7376\) | |
| 4200.bb6 | 4200o6 | \([0, 1, 0, 27992, -3986512]\) | \(75798394558/259416045\) | \(-8301313440000000\) | \([2]\) | \(24576\) | \(1.7376\) |
Rank
sage: E.rank()
The elliptic curves in class 4200o have rank \(1\).
Complex multiplication
The elliptic curves in class 4200o do not have complex multiplication.Modular form 4200.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
