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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 343850cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
343850.cf2 | 343850cf1 | \([1, 0, 0, -1726138, 1103537892]\) | \(-9836106385/3407872\) | \(-197065375436800000000\) | \([]\) | \(21384000\) | \(2.6055\) | \(\Gamma_0(N)\)-optimal |
343850.cf1 | 343850cf2 | \([1, 0, 0, -149846138, 706006617892]\) | \(-6434774386429585/140608\) | \(-8130871203325000000\) | \([]\) | \(64152000\) | \(3.1549\) |
Rank
sage: E.rank()
The elliptic curves in class 343850cf have rank \(1\).
Complex multiplication
The elliptic curves in class 343850cf do not have complex multiplication.Modular form 343850.2.a.cf
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 Cremona 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 Cremona labels.