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SageMath
E = EllipticCurve("qm1")
E.isogeny_class()
Elliptic curves in class 187200.qm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.qm1 | 187200gl2 | \([0, 0, 0, -6526380, -6417363760]\) | \(-6434774386429585/140608\) | \(-671765181235200\) | \([]\) | \(4976640\) | \(2.3714\) | |
187200.qm2 | 187200gl1 | \([0, 0, 0, -75180, -10031920]\) | \(-9836106385/3407872\) | \(-16281362025676800\) | \([]\) | \(1658880\) | \(1.8221\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.qm have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.qm do not have complex multiplication.Modular form 187200.2.a.qm
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.