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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6930.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.p1 | 6930q3 | \([1, -1, 0, -218349, 39325905]\) | \(1579250141304807889/41926500\) | \(30564418500\) | \([6]\) | \(41472\) | \(1.5260\) | |
| 6930.p2 | 6930q4 | \([1, -1, 0, -218079, 39427803]\) | \(-1573398910560073969/8138108343750\) | \(-5932680982593750\) | \([6]\) | \(82944\) | \(1.8726\) | |
| 6930.p3 | 6930q1 | \([1, -1, 0, -2889, 46413]\) | \(3658671062929/880165440\) | \(641640605760\) | \([2]\) | \(13824\) | \(0.97671\) | \(\Gamma_0(N)\)-optimal |
| 6930.p4 | 6930q2 | \([1, -1, 0, 6831, 285525]\) | \(48351870250991/76871856600\) | \(-56039583461400\) | \([2]\) | \(27648\) | \(1.3233\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.p have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.p do not have complex multiplication.Modular form 6930.2.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.
