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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 76050.cu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76050.cu1 | 76050ct2 | \([1, -1, 0, -127878867, 556722719541]\) | \(-168256703745625/30371328\) | \(-41745672231751800000000\) | \([]\) | \(13063680\) | \(3.3442\) | |
76050.cu2 | 76050ct1 | \([1, -1, 0, 455508, 2549221416]\) | \(7604375/2047032\) | \(-2813664483815371875000\) | \([]\) | \(4354560\) | \(2.7949\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.cu have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.cu do not have complex multiplication.Modular form 76050.2.a.cu
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.