Properties

Label 1680r
Number of curves $4$
Conductor $1680$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1680r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.o3 1680r1 [0, 1, 0, -56, 84] [2] 384 \(\Gamma_0(N)\)-optimal
1680.o2 1680r2 [0, 1, 0, -376, -2860] [2, 2] 768  
1680.o1 1680r3 [0, 1, 0, -5976, -179820] [2] 1536  
1680.o4 1680r4 [0, 1, 0, 104, -9196] [2] 1536  

Rank

sage: E.rank()
 

The elliptic curves in class 1680r have rank \(0\).

Complex multiplication

The elliptic curves in class 1680r do not have complex multiplication.

Modular form 1680.2.a.r

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{5} + q^{7} + q^{9} + 4q^{11} - 2q^{13} - q^{15} - 6q^{17} + O(q^{20}) \)

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.