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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4200a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.c3 | 4200a1 | \([0, -1, 0, -908, -10188]\) | \(20720464/105\) | \(420000000\) | \([2]\) | \(1536\) | \(0.50120\) | \(\Gamma_0(N)\)-optimal |
| 4200.c2 | 4200a2 | \([0, -1, 0, -1408, 2812]\) | \(19307236/11025\) | \(176400000000\) | \([2, 2]\) | \(3072\) | \(0.84777\) | |
| 4200.c1 | 4200a3 | \([0, -1, 0, -16408, 812812]\) | \(15267472418/36015\) | \(1152480000000\) | \([2]\) | \(6144\) | \(1.1943\) | |
| 4200.c4 | 4200a4 | \([0, -1, 0, 5592, 16812]\) | \(604223422/354375\) | \(-11340000000000\) | \([2]\) | \(6144\) | \(1.1943\) |
Rank
sage: E.rank()
The elliptic curves in class 4200a have rank \(1\).
Complex multiplication
The elliptic curves in class 4200a do not have complex multiplication.Modular form 4200.2.a.a
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.
