Properties

Label 328560.r
Number of curves $4$
Conductor $328560$
CM no
Rank $1$
Graph

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

Elliptic curves in class 328560.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
328560.r1 328560r3 \([0, -1, 0, -7019323440, 46602817329600]\) \(3639478711331685826729/2016912141902025000\) \(21196164084165759111488409600000\) \([2]\) \(756449280\) \(4.7014\)  
328560.r2 328560r2 \([0, -1, 0, -4281323440, -107187357070400]\) \(825824067562227826729/5613755625000000\) \(58996166904769743360000000000\) \([2, 2]\) \(378224640\) \(4.3548\)  
328560.r3 328560r1 \([0, -1, 0, -4274314160, -107557822751808]\) \(821774646379511057449/38361600000\) \(403150316386281062400000\) \([2]\) \(189112320\) \(4.0082\) \(\Gamma_0(N)\)-optimal
328560.r4 328560r4 \([0, -1, 0, -1655471920, -237267840008768]\) \(-47744008200656797609/2286529541015625000\) \(-24029631399290625000000000000000\) \([4]\) \(756449280\) \(4.7014\)  

Rank

sage: E.rank()
 

The elliptic curves in class 328560.r have rank \(1\).

Complex multiplication

The elliptic curves in class 328560.r do not have complex multiplication.

Modular form 328560.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.