Properties

Label 428910.w
Number of curves $4$
Conductor $428910$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 428910.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428910.w1 428910w4 \([1, 0, 1, -5500999, -4966434334]\) \(30949975477232209/478125000\) \(284399900353125000\) \([2]\) \(19267584\) \(2.4842\)  
428910.w2 428910w2 \([1, 0, 1, -354079, -72742798]\) \(8253429989329/936360000\) \(556968764851560000\) \([2, 2]\) \(9633792\) \(2.1376\)  
428910.w3 428910w1 \([1, 0, 1, -84959, 8316146]\) \(114013572049/15667200\) \(9319215934771200\) \([2]\) \(4816896\) \(1.7910\) \(\Gamma_0(N)\)-optimal*
428910.w4 428910w3 \([1, 0, 1, 486921, -365747198]\) \(21464092074671/109596256200\) \(-65190409082050840200\) \([2]\) \(19267584\) \(2.4842\)  
*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 428910.w1.

Rank

sage: E.rank()
 

The elliptic curves in class 428910.w have rank \(0\).

Complex multiplication

The elliptic curves in class 428910.w do not have complex multiplication.

Modular form 428910.2.a.w

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} - 2 q^{13} + 4 q^{14} - q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.