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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 67830.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67830.q1 | 67830t4 | \([1, 0, 1, -578294, 113832776]\) | \(21387926929424202058969/6774718645013256000\) | \(6774718645013256000\) | \([2]\) | \(1741824\) | \(2.3172\) | |
67830.q2 | 67830t2 | \([1, 0, 1, -523979, 145944722]\) | \(15909779688753017303209/1267090311060\) | \(1267090311060\) | \([6]\) | \(580608\) | \(1.7679\) | |
67830.q3 | 67830t1 | \([1, 0, 1, -32679, 2288602]\) | \(-3859320826323964009/34671204111600\) | \(-34671204111600\) | \([6]\) | \(290304\) | \(1.4214\) | \(\Gamma_0(N)\)-optimal |
67830.q4 | 67830t3 | \([1, 0, 1, 101706, 12104776]\) | \(116350853426302261031/130543121856000000\) | \(-130543121856000000\) | \([2]\) | \(870912\) | \(1.9707\) |
Rank
sage: E.rank()
The elliptic curves in class 67830.q have rank \(1\).
Complex multiplication
The elliptic curves in class 67830.q do not have complex multiplication.Modular form 67830.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.