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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 16731m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16731.a3 | 16731m1 | \([0, 0, 1, -507, -10436]\) | \(-4096/11\) | \(-38706181371\) | \([]\) | \(12960\) | \(0.71905\) | \(\Gamma_0(N)\)-optimal |
16731.a2 | 16731m2 | \([0, 0, 1, -15717, 1373674]\) | \(-122023936/161051\) | \(-566697201452811\) | \([]\) | \(64800\) | \(1.5238\) | |
16731.a1 | 16731m3 | \([0, 0, 1, -11894727, 15789916444]\) | \(-52893159101157376/11\) | \(-38706181371\) | \([]\) | \(324000\) | \(2.3285\) |
Rank
sage: E.rank()
The elliptic curves in class 16731m have rank \(1\).
Complex multiplication
The elliptic curves in class 16731m do not have complex multiplication.Modular form 16731.2.a.m
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.