Properties

Label 388080km
Number of curves $6$
Conductor $388080$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("km1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 388080km

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388080.km6 388080km1 [0, 0, 0, 246813, 13709786146] [2] 23592960 \(\Gamma_0(N)\)-optimal
388080.km5 388080km2 [0, 0, 0, -84460467, 293464049074] [2, 2] 47185920  
388080.km2 388080km3 [0, 0, 0, -1344697347, 18979500363586] [2, 2] 94371840  
388080.km4 388080km4 [0, 0, 0, -179540067, -488299438046] [2] 94371840  
388080.km1 388080km5 [0, 0, 0, -21515155347, 1214688284695186] [4] 188743680  
388080.km3 388080km6 [0, 0, 0, -1338029427, 19177040160754] [2] 188743680  

Rank

sage: E.rank()
 

The elliptic curves in class 388080km have rank \(1\).

Complex multiplication

The elliptic curves in class 388080km do not have complex multiplication.

Modular form 388080.2.a.km

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.