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SageMath
E = EllipticCurve("hq1")
E.isogeny_class()
Elliptic curves in class 117600.hq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.hq1 | 117600de4 | \([0, 1, 0, -125023500408, 17015100987713688]\) | \(229625675762164624948320008/9568125\) | \(9005442705000000000\) | \([2]\) | \(247726080\) | \(4.5377\) | |
117600.hq2 | 117600de3 | \([0, 1, 0, -7840766033, 263943665448063]\) | \(7079962908642659949376/100085966990454375\) | \(753600891549437872920000000000\) | \([2]\) | \(247726080\) | \(4.5377\) | |
117600.hq3 | 117600de1 | \([0, 1, 0, -7813969158, 265858972088688]\) | \(448487713888272974160064/91549016015625\) | \(10770650185222265625000000\) | \([2, 2]\) | \(123863040\) | \(4.1911\) | \(\Gamma_0(N)\)-optimal |
117600.hq4 | 117600de2 | \([0, 1, 0, -7787178408, 267772528198188]\) | \(-55486311952875723077768/801237030029296875\) | \(-754117882767333984375000000000\) | \([2]\) | \(247726080\) | \(4.5377\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.hq have rank \(1\).
Complex multiplication
The elliptic curves in class 117600.hq do not have complex multiplication.Modular form 117600.2.a.hq
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.