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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 199920.ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
199920.ct1 | 199920cg2 | \([0, -1, 0, -29543880, -61721933328]\) | \(5918043195362419129/8515734343200\) | \(4103649811427888332800\) | \([2]\) | \(17694720\) | \(3.0496\) | |
199920.ct2 | 199920cg1 | \([0, -1, 0, -1319880, -1525786128]\) | \(-527690404915129/1782829440000\) | \(-859128220821749760000\) | \([2]\) | \(8847360\) | \(2.7030\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 199920.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 199920.ct do not have complex multiplication.Modular form 199920.2.a.ct
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.