Properties

Label 424830.ff
Number of curves $4$
Conductor $424830$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ff1")
 
E.isogeny_class()
 

Elliptic curves in class 424830.ff

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.ff1 424830ff3 \([1, 1, 1, -92627396, 343085724029]\) \(30949975477232209/478125000\) \(1357760658931228125000\) \([2]\) \(63700992\) \(3.1901\) \(\Gamma_0(N)\)-optimal*
424830.ff2 424830ff2 \([1, 1, 1, -5962076, 5021643773]\) \(8253429989329/936360000\) \(2659038474450917160000\) \([2, 2]\) \(31850496\) \(2.8435\) \(\Gamma_0(N)\)-optimal*
424830.ff3 424830ff1 \([1, 1, 1, -1430556, -575689731]\) \(114013572049/15667200\) \(44491101271858483200\) \([2]\) \(15925248\) \(2.4969\) \(\Gamma_0(N)\)-optimal*
424830.ff4 424830ff4 \([1, 1, 1, 8198924, 25277538173]\) \(21464092074671/109596256200\) \(-311227158242107598992200\) \([2]\) \(63700992\) \(3.1901\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 424830.ff1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830.ff have rank \(1\).

Complex multiplication

The elliptic curves in class 424830.ff do not have complex multiplication.

Modular form 424830.2.a.ff

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + q^{15} + q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.