Properties

Label 4032.d
Number of curves $2$
Conductor $4032$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4032.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.d1 4032p2 \([0, 0, 0, -1452, 21040]\) \(3543122/49\) \(4682022912\) \([2]\) \(3072\) \(0.66121\)  
4032.d2 4032p1 \([0, 0, 0, -12, 880]\) \(-4/7\) \(-334430208\) \([2]\) \(1536\) \(0.31464\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 4032.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.