Properties

Label 471510.bo
Number of curves $2$
Conductor $471510$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bo1")
 
E.isogeny_class()
 

Elliptic curves in class 471510.bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
471510.bo1 471510bo2 \([1, -1, 0, -332258679, -2331026111907]\) \(1152829477932246539641/3188367360\) \(11219047755776040960\) \([2]\) \(76677120\) \(3.3135\) \(\Gamma_0(N)\)-optimal*
471510.bo2 471510bo1 \([1, -1, 0, -20757879, -36448918947]\) \(-281115640967896441/468084326400\) \(-1647068803141887590400\) \([2]\) \(38338560\) \(2.9669\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 471510.bo1.

Rank

sage: E.rank()
 

The elliptic curves in class 471510.bo have rank \(1\).

Complex multiplication

The elliptic curves in class 471510.bo do not have complex multiplication.

Modular form 471510.2.a.bo

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - 4 q^{7} - q^{8} - q^{10} + 2 q^{11} + 4 q^{14} + q^{16} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.