Properties

Label 4004.c
Number of curves $2$
Conductor $4004$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 4004.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4004.c1 4004c1 \([0, -1, 0, -21, 38]\) \(67108864/13013\) \(208208\) \([2]\) \(840\) \(-0.26250\) \(\Gamma_0(N)\)-optimal
4004.c2 4004c2 \([0, -1, 0, 44, 168]\) \(35969456/77077\) \(-19731712\) \([2]\) \(1680\) \(0.084069\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4004.c have rank \(0\).

Complex multiplication

The elliptic curves in class 4004.c do not have complex multiplication.

Modular form 4004.2.a.c

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} + 4 q^{5} + q^{7} + q^{9} + q^{11} - q^{13} + 8 q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.