Properties

Label 400078a
Number of curves $2$
Conductor $400078$
CM no
Rank $0$
Graph

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

Elliptic curves in class 400078a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400078.a2 400078a1 \([1, 0, 1, 52916, -904566]\) \(3449795831/2071552\) \(-9840087940652032\) \([2]\) \(5529600\) \(1.7580\) \(\Gamma_0(N)\)-optimal*
400078.a1 400078a2 \([1, 0, 1, -216044, -7359606]\) \(234770924809/130960928\) \(622078059498095648\) \([2]\) \(11059200\) \(2.1046\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 400078a1.

Rank

sage: E.rank()
 

The elliptic curves in class 400078a have rank \(0\).

Complex multiplication

The elliptic curves in class 400078a do not have complex multiplication.

Modular form 400078.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.