Properties

Label 4032.bc
Number of curves $4$
Conductor $4032$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 4032.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.bc1 4032bd4 \([0, 0, 0, -145164, 21288080]\) \(7080974546692/189\) \(9029615616\) \([2]\) \(12288\) \(1.4241\)  
4032.bc2 4032bd3 \([0, 0, 0, -14124, -77488]\) \(6522128932/3720087\) \(177729924169728\) \([2]\) \(12288\) \(1.4241\)  
4032.bc3 4032bd2 \([0, 0, 0, -9084, 331760]\) \(6940769488/35721\) \(426649337856\) \([2, 2]\) \(6144\) \(1.0775\)  
4032.bc4 4032bd1 \([0, 0, 0, -264, 10712]\) \(-2725888/64827\) \(-48393096192\) \([2]\) \(3072\) \(0.73092\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 4032.bc have rank \(1\).

Complex multiplication

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

Modular form 4032.2.a.bc

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