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SageMath
E = EllipticCurve("qg1")
E.isogeny_class()
Elliptic curves in class 187200.qg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.qg1 | 187200kn2 | \([0, 0, 0, -163159500, 802170470000]\) | \(-6434774386429585/140608\) | \(-10496330956800000000\) | \([]\) | \(24883200\) | \(3.1761\) | |
187200.qg2 | 187200kn1 | \([0, 0, 0, -1879500, 1253990000]\) | \(-9836106385/3407872\) | \(-254396281651200000000\) | \([]\) | \(8294400\) | \(2.6268\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.qg have rank \(1\).
Complex multiplication
The elliptic curves in class 187200.qg do not have complex multiplication.Modular form 187200.2.a.qg
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.