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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 129600.ir
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129600.ir1 | 129600cc2 | \([0, 0, 0, -89100, -11178000]\) | \(-35937/4\) | \(-8707129344000000\) | \([]\) | \(746496\) | \(1.7959\) | |
129600.ir2 | 129600cc1 | \([0, 0, 0, 6900, 22000]\) | \(109503/64\) | \(-21233664000000\) | \([]\) | \(248832\) | \(1.2466\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129600.ir have rank \(1\).
Complex multiplication
The elliptic curves in class 129600.ir do not have complex multiplication.Modular form 129600.2.a.ir
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.