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Properties

Label 478800.z
Number of curves $4$
Conductor $478800$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 478800.z have rank \(2\).

Complex multiplication

The elliptic curves in class 478800.z do not have complex multiplication.

Modular form 478800.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q - q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} - q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 478800.z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
478800.z1 478800z4 \([0, 0, 0, -1035075, -404414750]\) \(5257286722802/13683705\) \(319213470240000000\) \([2]\) \(7077888\) \(2.2343\)  
478800.z2 478800z3 \([0, 0, 0, -945075, 352215250]\) \(4001704635602/18475695\) \(431001012960000000\) \([2]\) \(7077888\) \(2.2343\) \(\Gamma_0(N)\)-optimal*
478800.z3 478800z2 \([0, 0, 0, -90075, -899750]\) \(6929294404/3980025\) \(46423011600000000\) \([2, 2]\) \(3538944\) \(1.8877\) \(\Gamma_0(N)\)-optimal*
478800.z4 478800z1 \([0, 0, 0, 22425, -112250]\) \(427694384/249375\) \(-727177500000000\) \([2]\) \(1769472\) \(1.5411\) \(\Gamma_0(N)\)-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 478800.z1.