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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 4200.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.t1 | 4200x4 | \([0, 1, 0, -100808, 12285888]\) | \(7080974546692/189\) | \(3024000000\) | \([2]\) | \(12288\) | \(1.3329\) | |
| 4200.t2 | 4200x3 | \([0, 1, 0, -9808, -48112]\) | \(6522128932/3720087\) | \(59521392000000\) | \([2]\) | \(12288\) | \(1.3329\) | |
| 4200.t3 | 4200x2 | \([0, 1, 0, -6308, 189888]\) | \(6940769488/35721\) | \(142884000000\) | \([2, 2]\) | \(6144\) | \(0.98633\) | |
| 4200.t4 | 4200x1 | \([0, 1, 0, -183, 6138]\) | \(-2725888/64827\) | \(-16206750000\) | \([2]\) | \(3072\) | \(0.63976\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4200.t have rank \(1\).
Complex multiplication
The elliptic curves in class 4200.t do not have complex multiplication.Modular form 4200.2.a.t
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
