Properties

Label 212160gf
Number of curves 8
Conductor 212160
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("212160.cc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212160gf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
212160.cc7 212160gf1 [0, -1, 0, -134153345, 575935024257] [2] 53084160 \(\Gamma_0(N)\)-optimal
212160.cc6 212160gf2 [0, -1, 0, -355665025, -1815815189375] [2, 2] 106168320  
212160.cc5 212160gf3 [0, -1, 0, -1650738305, -25645921710975] [2] 159252480  
212160.cc8 212160gf4 [0, -1, 0, 952372095, -12070041387903] [2] 212336640  
212160.cc4 212160gf5 [0, -1, 0, -5207889025, -144637146850175] [2] 212336640  
212160.cc2 212160gf6 [0, -1, 0, -26364000385, -1647641566502783] [2, 2] 318504960  
212160.cc3 212160gf7 [0, -1, 0, -26316193665, -1653914697371775] [2] 637009920  
212160.cc1 212160gf8 [0, -1, 0, -421824000385, -105449485114502783] [2] 637009920  

Rank

sage: E.rank()
 

The elliptic curves in class 212160gf have rank \(1\).

Modular form 212160.2.a.cc

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.