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Properties

Label 457776.fr
Number of curves $2$
Conductor $457776$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 457776.fr have rank \(0\).

Complex multiplication

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

Modular form 457776.2.a.fr

Copy content sage:E.q_eigenform(10)
 
\(q + 3 q^{5} + 4 q^{7} - q^{11} + 2 q^{13} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 457776.fr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
457776.fr1 457776fr2 \([0, 0, 0, -579891216, -5374869341668]\) \(-16565495781326848/107811\) \(-140353077250261396224\) \([]\) \(101523456\) \(3.4691\)  
457776.fr2 457776fr1 \([0, 0, 0, -6838896, -8062449172]\) \(-27172077568/5845851\) \(-7610384626768305955584\) \([]\) \(33841152\) \(2.9198\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 457776.fr1.