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Properties

Label 457776.em
Number of curves $2$
Conductor $457776$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("em1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 457776.em have rank \(1\).

Complex multiplication

The elliptic curves in class 457776.em do not have complex multiplication.

Modular form 457776.2.a.em

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - 2 q^{7} - q^{11} + 4 q^{13} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 457776.em

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
457776.em1 457776em2 \([0, 0, 0, -46114981779, 2717051698367570]\) \(150476552140919246594353/42832838728685592576\) \(3087150930345044547213414258180096\) \([2]\) \(1794244608\) \(5.1327\) \(\Gamma_0(N)\)-optimal*
457776.em2 457776em1 \([0, 0, 0, -17136928659, -829960528901038]\) \(7722211175253055152433/340131399900069888\) \(24514764811463280186416481435648\) \([2]\) \(897122304\) \(4.7861\) \(\Gamma_0(N)\)-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 457776.em1.