Properties

Label 4032f
Number of curves $4$
Conductor $4032$
CM no
Rank $0$
Graph

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

Elliptic curves in class 4032f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.be3 4032f1 \([0, 0, 0, -264, 1640]\) \(2725888/21\) \(15676416\) \([2]\) \(1024\) \(0.20969\) \(\Gamma_0(N)\)-optimal
4032.be2 4032f2 \([0, 0, 0, -444, -880]\) \(810448/441\) \(5267275776\) \([2, 2]\) \(2048\) \(0.55626\)  
4032.be1 4032f3 \([0, 0, 0, -5484, -156112]\) \(381775972/567\) \(27088846848\) \([2]\) \(4096\) \(0.90284\)  
4032.be4 4032f4 \([0, 0, 0, 1716, -6928]\) \(11696828/7203\) \(-344128684032\) \([2]\) \(4096\) \(0.90284\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4032f have rank \(0\).

Complex multiplication

The elliptic curves in class 4032f do not have complex multiplication.

Modular form 4032.2.a.f

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} - q^{7} + 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.