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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 187200.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.bc1 | 187200co2 | \([0, 0, 0, -1937100, -1037872240]\) | \(-168256703745625/30371328\) | \(-145101279146803200\) | \([]\) | \(2985984\) | \(2.2967\) | |
187200.bc2 | 187200co1 | \([0, 0, 0, 6900, -4752880]\) | \(7604375/2047032\) | \(-9779847679180800\) | \([]\) | \(995328\) | \(1.7474\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.bc do not have complex multiplication.Modular form 187200.2.a.bc
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.