Properties

Label 435344.z
Number of curves $4$
Conductor $435344$
CM no
Rank $0$
Graph

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

Elliptic curves in class 435344.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.z1 435344z4 \([0, 0, 0, -334451, -74429966]\) \(209267191953/55223\) \(1091792377475072\) \([2]\) \(2457600\) \(1.8700\)  
435344.z2 435344z2 \([0, 0, 0, -23491, -856830]\) \(72511713/25921\) \(512473973100544\) \([2, 2]\) \(1228800\) \(1.5234\)  
435344.z3 435344z1 \([0, 0, 0, -9971, 373490]\) \(5545233/161\) \(3183068155904\) \([2]\) \(614400\) \(1.1768\) \(\Gamma_0(N)\)-optimal*
435344.z4 435344z3 \([0, 0, 0, 71149, -6024174]\) \(2014698447/1958887\) \(-38728390252883968\) \([2]\) \(2457600\) \(1.8700\)  
*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 435344.z1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344.z have rank \(0\).

Complex multiplication

The elliptic curves in class 435344.z do not have complex multiplication.

Modular form 435344.2.a.z

sage: E.q_eigenform(10)
 
\(q - 2q^{5} + q^{7} - 3q^{9} + 4q^{11} - 2q^{17} + 4q^{19} + O(q^{20})\)  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.