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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 193200.cx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.cx1 | 193200ea2 | \([0, -1, 0, -295008, 208512]\) | \(44365623586201/25674468750\) | \(1643166000000000000\) | \([2]\) | \(2654208\) | \(2.1847\) | |
193200.cx2 | 193200ea1 | \([0, -1, 0, -203008, 35168512]\) | \(14457238157881/49990500\) | \(3199392000000000\) | \([2]\) | \(1327104\) | \(1.8382\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.cx have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.cx do not have complex multiplication.Modular form 193200.2.a.cx
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.