Properties

Label 78400.ey
Number of curves $4$
Conductor $78400$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ey1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 78400.ey

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78400.ey1 78400gz4 \([0, 0, 0, -1465100, 682570000]\) \(1443468546/7\) \(1686616064000000\) \([2]\) \(786432\) \(2.1222\)  
78400.ey2 78400gz3 \([0, 0, 0, -289100, -47334000]\) \(11090466/2401\) \(578509309952000000\) \([2]\) \(786432\) \(2.1222\)  
78400.ey3 78400gz2 \([0, 0, 0, -93100, 10290000]\) \(740772/49\) \(5903156224000000\) \([2, 2]\) \(393216\) \(1.7756\)  
78400.ey4 78400gz1 \([0, 0, 0, 4900, 686000]\) \(432/7\) \(-210827008000000\) \([2]\) \(196608\) \(1.4291\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 78400.ey have rank \(1\).

Complex multiplication

The elliptic curves in class 78400.ey do not have complex multiplication.

Modular form 78400.2.a.ey

sage: E.q_eigenform(10)
 
\(q - 3q^{9} - 4q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.