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Properties

Label 424830.fi
Number of curves $6$
Conductor $424830$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("fi1")
 
E.isogeny_class()
 

Elliptic curves in class 424830.fi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.fi1 424830fi5 \([1, 1, 1, -24676675675, 1492019722415585]\) \(585196747116290735872321/836876053125000\) \(2376527856386437592803125000\) \([2]\) \(1019215872\) \(4.5242\) \(\Gamma_0(N)\)-optimal*
424830.fi2 424830fi4 \([1, 1, 1, -3577352115, -82343521821135]\) \(1782900110862842086081/328139630024640\) \(931838076410362593504123840\) \([2]\) \(509607936\) \(4.1777\)  
424830.fi3 424830fi3 \([1, 1, 1, -1556294195, 22867449803057]\) \(146796951366228945601/5397929064360000\) \(15328827656553121694885160000\) \([2, 2]\) \(509607936\) \(4.1777\) \(\Gamma_0(N)\)-optimal*
424830.fi4 424830fi2 \([1, 1, 1, -246684915, -1004631996495]\) \(584614687782041281/184812061593600\) \(524822058097286387835801600\) \([2, 2]\) \(254803968\) \(3.8311\) \(\Gamma_0(N)\)-optimal*
424830.fi5 424830fi1 \([1, 1, 1, 43332365, -106622490703]\) \(3168685387909439/3563732336640\) \(-10120147588289363013795840\) \([2]\) \(127401984\) \(3.4845\) \(\Gamma_0(N)\)-optimal*
424830.fi6 424830fi6 \([1, 1, 1, 610338805, 81554604547457]\) \(8854313460877886399/1016927675429790600\) \(-2887831405337421323522734158600\) \([2]\) \(1019215872\) \(4.5242\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 424830.fi1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830.fi have rank \(0\).

Complex multiplication

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

Modular form 424830.2.a.fi

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.