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Properties

Label 3234.b
Number of curves $2$
Conductor $3234$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 3234.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.b1 3234c2 \([1, 1, 0, -22159540, -39589128368]\) \(10228636028672744397625/167006381634183168\) \(19648133792880015532032\) \([2]\) \(399360\) \(3.0769\)  
3234.b2 3234c1 \([1, 1, 0, -82100, -1735149744]\) \(-520203426765625/11054534935707648\) \(-1300554980651069079552\) \([2]\) \(199680\) \(2.7303\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3234.2.a.b

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