Properties

Label 78a
Number of curves $4$
Conductor $78$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 78a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78.a4 78a1 \([1, 1, 0, -19, 685]\) \(-822656953/207028224\) \(-207028224\) \([2]\) \(40\) \(0.27493\) \(\Gamma_0(N)\)-optimal
78.a3 78a2 \([1, 1, 0, -1299, 17325]\) \(242702053576633/2554695936\) \(2554695936\) \([2, 2]\) \(80\) \(0.62150\)  
78.a2 78a3 \([1, 1, 0, -2339, -15747]\) \(1416134368422073/725251155408\) \(725251155408\) \([2]\) \(160\) \(0.96808\)  
78.a1 78a4 \([1, 1, 0, -20739, 1140957]\) \(986551739719628473/111045168\) \(111045168\) \([4]\) \(160\) \(0.96808\)  

Rank

sage: E.rank()
 

The elliptic curves in class 78a have rank \(0\).

Complex multiplication

The elliptic curves in class 78a do not have complex multiplication.

Modular form 78.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} + q^{13} - 4q^{14} - 2q^{15} + q^{16} + 2q^{17} - q^{18} - 8q^{19} + O(q^{20})\)  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 Cremona 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 Cremona labels.