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Properties

Label 3234.u
Number of curves $2$
Conductor $3234$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 3234.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.u1 3234v2 \([1, 0, 0, -52459, 4620209]\) \(46546832455691959/748268928\) \(256656242304\) \([2]\) \(10752\) \(1.3221\)  
3234.u2 3234v1 \([1, 0, 0, -3179, 76593]\) \(-10358806345399/1445216256\) \(-495709175808\) \([2]\) \(5376\) \(0.97555\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3234.u have rank \(1\).

Complex multiplication

The elliptic curves in class 3234.u do not have complex multiplication.

Modular form 3234.2.a.u

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