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Properties

Label 408135bo
Number of curves $2$
Conductor $408135$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("bo1")
 
E.isogeny_class()
 

Elliptic curves in class 408135bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
408135.bo1 408135bo1 \([1, 0, 1, -38874, -2538209]\) \(1345938541921/203765625\) \(983537752640625\) \([2]\) \(1658880\) \(1.6008\) \(\Gamma_0(N)\)-optimal
408135.bo2 408135bo2 \([1, 0, 1, 66751, -13903459]\) \(6814692748079/21258460125\) \(-102610526657491125\) \([2]\) \(3317760\) \(1.9474\)  

Rank

sage: E.rank()
 

The elliptic curves in class 408135bo have rank \(1\).

Complex multiplication

The elliptic curves in class 408135bo do not have complex multiplication.

Modular form 408135.2.a.bo

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{10} + 2 q^{11} - q^{12} - q^{14} - q^{15} - q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.