Properties

Label 3234.p
Number of curves $4$
Conductor $3234$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 3234.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.p1 3234r3 \([1, 1, 1, -1972104, -1066786449]\) \(7209828390823479793/49509306\) \(5824720341594\) \([2]\) \(36864\) \(2.0492\)  
3234.p2 3234r4 \([1, 1, 1, -171844, -2403745]\) \(4770223741048753/2740574865798\) \(322425892386268902\) \([2]\) \(36864\) \(2.0492\)  
3234.p3 3234r2 \([1, 1, 1, -123334, -16685089]\) \(1763535241378513/4612311396\) \(542633823428004\) \([2, 2]\) \(18432\) \(1.7026\)  
3234.p4 3234r1 \([1, 1, 1, -4754, -463345]\) \(-100999381393/723148272\) \(-85077671052528\) \([4]\) \(9216\) \(1.3560\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3234.p have rank \(0\).

Complex multiplication

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

Modular form 3234.2.a.p

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.