Properties

Label 42432.r
Number of curves $2$
Conductor $42432$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 42432.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42432.r1 42432k2 \([0, -1, 0, -6193, 86833]\) \(1603530178000/738501777\) \(12099613114368\) \([2]\) \(73728\) \(1.2051\)  
42432.r2 42432k1 \([0, -1, 0, -3133, -65555]\) \(3322336000000/51429573\) \(52663882752\) \([2]\) \(36864\) \(0.85852\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42432.r have rank \(1\).

Complex multiplication

The elliptic curves in class 42432.r do not have complex multiplication.

Modular form 42432.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{7} + q^{9} + 2 q^{11} + q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.