Properties

Label 476850y
Number of curves $2$
Conductor $476850$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 476850y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
476850.y2 476850y1 \([1, 1, 0, 5974925, -45266750375]\) \(101847563/3881196\) \(-898950765446582835937500\) \([2]\) \(81469440\) \(3.2762\) \(\Gamma_0(N)\)-optimal*
476850.y1 476850y2 \([1, 1, 0, -159838825, -744171706625]\) \(1949845587157/95664294\) \(22157471644618551011718750\) \([2]\) \(162938880\) \(3.6227\) \(\Gamma_0(N)\)-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 476850y1.

Rank

sage: E.rank()
 

The elliptic curves in class 476850y have rank \(1\).

Complex multiplication

The elliptic curves in class 476850y do not have complex multiplication.

Modular form 476850.2.a.y

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.