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Properties

Label 17670bb
Number of curves $3$
Conductor $17670$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 17670bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17670.z3 17670bb1 \([1, 0, 0, -1469295, 666308025]\) \(350792849898814825511281/11141148807000000000\) \(11141148807000000000\) \([9]\) \(427680\) \(2.4293\) \(\Gamma_0(N)\)-optimal
17670.z2 17670bb2 \([1, 0, 0, -16184295, -24834624975]\) \(468818856965932972707671281/4896432946801144503000\) \(4896432946801144503000\) \([3]\) \(1283040\) \(2.9786\)  
17670.z1 17670bb3 \([1, 0, 0, -1307624145, -18200156001645]\) \(247270613043280364880287393857681/288395676136025670\) \(288395676136025670\) \([]\) \(3849120\) \(3.5279\)  

Rank

sage: E.rank()
 

The elliptic curves in class 17670bb have rank \(0\).

Complex multiplication

The elliptic curves in class 17670bb do not have complex multiplication.

Modular form 17670.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + 3 q^{11} + q^{12} + 5 q^{13} - q^{14} + q^{15} + q^{16} + q^{18} + q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.