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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 134640bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
134640.ft4 | 134640bm1 | \([0, 0, 0, -23907, 7938146]\) | \(-506071034209/8823767040\) | \(-26347627201167360\) | \([2]\) | \(983040\) | \(1.8320\) | \(\Gamma_0(N)\)-optimal |
134640.ft3 | 134640bm2 | \([0, 0, 0, -761187, 254632034]\) | \(16334668434139489/72511718400\) | \(216518830954905600\) | \([2, 2]\) | \(1966080\) | \(2.1786\) | |
134640.ft1 | 134640bm3 | \([0, 0, 0, -12165987, 16333119074]\) | \(66692696957462376289/1322972640\) | \(3950375135477760\) | \([2]\) | \(3932160\) | \(2.5251\) | |
134640.ft2 | 134640bm4 | \([0, 0, 0, -1152867, -35446174]\) | \(56751044592329569/32660264340000\) | \(97523026755010560000\) | \([2]\) | \(3932160\) | \(2.5251\) |
Rank
sage: E.rank()
The elliptic curves in class 134640bm have rank \(1\).
Complex multiplication
The elliptic curves in class 134640bm do not have complex multiplication.Modular form 134640.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.