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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6006.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6006.p1 | 6006r4 | \([1, 0, 1, -43401, -3482108]\) | \(9040834853442015625/4807849294248\) | \(4807849294248\) | \([2]\) | \(20736\) | \(1.3832\) | |
| 6006.p2 | 6006r3 | \([1, 0, 1, -2241, -74060]\) | \(-1243857621903625/1637696668608\) | \(-1637696668608\) | \([2]\) | \(10368\) | \(1.0366\) | |
| 6006.p3 | 6006r2 | \([1, 0, 1, -1656, 20116]\) | \(501796540869625/113170859802\) | \(113170859802\) | \([6]\) | \(6912\) | \(0.83385\) | |
| 6006.p4 | 6006r1 | \([1, 0, 1, 234, 1972]\) | \(1425727406375/2472321852\) | \(-2472321852\) | \([6]\) | \(3456\) | \(0.48728\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6006.p have rank \(1\).
Complex multiplication
The elliptic curves in class 6006.p do not have complex multiplication.Modular form 6006.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
