Properties

Label 388080fd
Number of curves $4$
Conductor $388080$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 388080fd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388080.fd4 388080fd1 \([0, 0, 0, 96432, 42349867]\) \(72268906496/606436875\) \(-832187814401790000\) \([2]\) \(3317760\) \(2.1199\) \(\Gamma_0(N)\)-optimal
388080.fd3 388080fd2 \([0, 0, 0, -1391943, 582034642]\) \(13584145739344/1195803675\) \(26255217326667436800\) \([2]\) \(6635520\) \(2.4665\)  
388080.fd2 388080fd3 \([0, 0, 0, -6889008, 6965095543]\) \(-26348629355659264/24169921875\) \(-33167367105468750000\) \([2]\) \(9953280\) \(2.6692\)  
388080.fd1 388080fd4 \([0, 0, 0, -110248383, 445560267418]\) \(6749703004355978704/5671875\) \(124532407692000000\) \([2]\) \(19906560\) \(3.0158\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 388080.2.a.fd

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.