Properties

Label 3234.q
Number of curves $4$
Conductor $3234$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3234.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.q1 3234p4 \([1, 1, 1, -692518, 221512829]\) \(312196988566716625/25367712678\) \(2984486028854022\) \([2]\) \(27648\) \(2.0145\)  
3234.q2 3234p3 \([1, 1, 1, -40328, 3942245]\) \(-61653281712625/21875235228\) \(-2573599549338972\) \([2]\) \(13824\) \(1.6679\)  
3234.q3 3234p2 \([1, 1, 1, -17788, -465403]\) \(5290763640625/2291573592\) \(269601341525208\) \([2]\) \(9216\) \(1.4652\)  
3234.q4 3234p1 \([1, 1, 1, 3772, -51451]\) \(50447927375/39517632\) \(-4649209887168\) \([2]\) \(4608\) \(1.1186\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3234.2.a.q

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.