Properties

Label 1600a
Number of curves $4$
Conductor $1600$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 1600a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1600.o3 1600a1 \([0, 0, 0, -200, 1000]\) \(55296/5\) \(80000000\) \([2]\) \(384\) \(0.25593\) \(\Gamma_0(N)\)-optimal
1600.o2 1600a2 \([0, 0, 0, -700, -6000]\) \(148176/25\) \(6400000000\) \([2, 2]\) \(768\) \(0.60250\)  
1600.o1 1600a3 \([0, 0, 0, -10700, -426000]\) \(132304644/5\) \(5120000000\) \([2]\) \(1536\) \(0.94908\)  
1600.o4 1600a4 \([0, 0, 0, 1300, -34000]\) \(237276/625\) \(-640000000000\) \([2]\) \(1536\) \(0.94908\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1600a have rank \(1\).

Complex multiplication

The elliptic curves in class 1600a do not have complex multiplication.

Modular form 1600.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.