Properties

Label 4032.e
Number of curves $6$
Conductor $4032$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 4032.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.e1 4032i3 \([0, 0, 0, -774156, -262174736]\) \(268498407453697/252\) \(48157949952\) \([2]\) \(24576\) \(1.7781\)  
4032.e2 4032i5 \([0, 0, 0, -526476, 145621744]\) \(84448510979617/933897762\) \(178470641597939712\) \([2]\) \(49152\) \(2.1247\)  
4032.e3 4032i4 \([0, 0, 0, -59916, -1997840]\) \(124475734657/63011844\) \(12041750911647744\) \([2, 2]\) \(24576\) \(1.7781\)  
4032.e4 4032i2 \([0, 0, 0, -48396, -4094480]\) \(65597103937/63504\) \(12135803387904\) \([2, 2]\) \(12288\) \(1.4316\)  
4032.e5 4032i1 \([0, 0, 0, -2316, -94736]\) \(-7189057/16128\) \(-3082108796928\) \([2]\) \(6144\) \(1.0850\) \(\Gamma_0(N)\)-optimal
4032.e6 4032i6 \([0, 0, 0, 222324, -15432464]\) \(6359387729183/4218578658\) \(-806182936033886208\) \([2]\) \(49152\) \(2.1247\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 4032.2.a.e

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - q^{7} - 4 q^{11} - 6 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.