Properties

Label 4032.w
Number of curves $6$
Conductor $4032$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4032.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.w1 4032l6 \([0, 0, 0, -1572780, 759189296]\) \(2251439055699625/25088\) \(4794391461888\) \([2]\) \(27648\) \(2.0021\)  
4032.w2 4032l5 \([0, 0, 0, -98220, 11882288]\) \(-548347731625/1835008\) \(-350675489783808\) \([2]\) \(13824\) \(1.6556\)  
4032.w3 4032l4 \([0, 0, 0, -20460, 923312]\) \(4956477625/941192\) \(179864592187392\) \([2]\) \(9216\) \(1.4528\)  
4032.w4 4032l2 \([0, 0, 0, -6060, -181456]\) \(128787625/98\) \(18728091648\) \([2]\) \(3072\) \(0.90352\)  
4032.w5 4032l1 \([0, 0, 0, -300, -4048]\) \(-15625/28\) \(-5350883328\) \([2]\) \(1536\) \(0.55694\) \(\Gamma_0(N)\)-optimal
4032.w6 4032l3 \([0, 0, 0, 2580, 84656]\) \(9938375/21952\) \(-4195092529152\) \([2]\) \(4608\) \(1.1062\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 4032.2.a.w

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.