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Properties

Label 459186bb
Number of curves $2$
Conductor $459186$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 459186bb have rank \(1\).

Complex multiplication

The elliptic curves in class 459186bb do not have complex multiplication.

Modular form 459186.2.a.bb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} + q^{12} + q^{13} - q^{14} - 3 q^{15} + q^{16} - 3 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 459186bb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.bb2 459186bb1 \([1, 0, 1, -77705895, 236352346090]\) \(103729341683356873/11784544559424\) \(5895176144230924108294464\) \([3]\) \(153342720\) \(3.4855\) \(\Gamma_0(N)\)-optimal
459186.bb1 459186bb2 \([1, 0, 1, -1458001350, -21393078204008]\) \(685193929756685418553/1302013270032384\) \(651327467961321972307329024\) \([]\) \(460028160\) \(4.0348\)