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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 162240.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162240.cd1 | 162240bz4 | \([0, -1, 0, -324705, -71108415]\) | \(23937672968/45\) | \(7117419479040\) | \([2]\) | \(1179648\) | \(1.7214\) | |
162240.cd2 | 162240bz3 | \([0, -1, 0, -54305, 3440865]\) | \(111980168/32805\) | \(5188598800220160\) | \([2]\) | \(1179648\) | \(1.7214\) | |
162240.cd3 | 162240bz2 | \([0, -1, 0, -20505, -1081575]\) | \(48228544/2025\) | \(40035484569600\) | \([2, 2]\) | \(589824\) | \(1.3748\) | |
162240.cd4 | 162240bz1 | \([0, -1, 0, 620, -63350]\) | \(85184/5625\) | \(-1737651240000\) | \([2]\) | \(294912\) | \(1.0282\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162240.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 162240.cd do not have complex multiplication.Modular form 162240.2.a.cd
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.