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SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 223440hg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223440.j4 | 223440hg1 | \([0, -1, 0, 964, 19776]\) | \(3286064/7695\) | \(-231759118080\) | \([2]\) | \(294912\) | \(0.86475\) | \(\Gamma_0(N)\)-optimal |
223440.j3 | 223440hg2 | \([0, -1, 0, -7856, 224400]\) | \(445138564/81225\) | \(9785384985600\) | \([2, 2]\) | \(589824\) | \(1.2113\) | |
223440.j1 | 223440hg3 | \([0, -1, 0, -119576, 15954576]\) | \(784767874322/35625\) | \(8583671040000\) | \([2]\) | \(1179648\) | \(1.5579\) | |
223440.j2 | 223440hg4 | \([0, -1, 0, -37256, -2550960]\) | \(23735908082/1954815\) | \(471003197306880\) | \([2]\) | \(1179648\) | \(1.5579\) |
Rank
sage: E.rank()
The elliptic curves in class 223440hg have rank \(2\).
Complex multiplication
The elliptic curves in class 223440hg do not have complex multiplication.Modular form 223440.2.a.hg
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.