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SageMath
E = EllipticCurve("gj1")
E.isogeny_class()
Elliptic curves in class 117600.gj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.gj1 | 117600cn4 | \([0, 1, 0, -17297408, 27683992188]\) | \(608119035935048/826875\) | \(778248135000000000\) | \([2]\) | \(3538944\) | \(2.7072\) | |
117600.gj2 | 117600cn3 | \([0, 1, 0, -2744408, -1169976312]\) | \(2428799546888/778248135\) | \(732480918676920000000\) | \([2]\) | \(3538944\) | \(2.7072\) | |
117600.gj3 | 117600cn1 | \([0, 1, 0, -1090658, 424238688]\) | \(1219555693504/43758225\) | \(5148111413025000000\) | \([2, 2]\) | \(1769472\) | \(2.3606\) | \(\Gamma_0(N)\)-optimal |
117600.gj4 | 117600cn2 | \([0, 1, 0, 409967, 1503188063]\) | \(1012048064/130203045\) | \(-980368514637120000000\) | \([2]\) | \(3538944\) | \(2.7072\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.gj have rank \(1\).
Complex multiplication
The elliptic curves in class 117600.gj do not have complex multiplication.Modular form 117600.2.a.gj
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.