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Properties

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

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 3234f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.f2 3234f1 \([1, 1, 0, -60, 288]\) \(-500313625/574992\) \(-28174608\) \([]\) \(864\) \(0.12372\) \(\Gamma_0(N)\)-optimal
3234.f1 3234f2 \([1, 1, 0, -5835, 169149]\) \(-448504189023625/135168\) \(-6623232\) \([]\) \(2592\) \(0.67303\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3234.2.a.f

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.