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Properties

Label 400752.bi
Number of curves $2$
Conductor $400752$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bi1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 400752.bi have rank \(1\).

Complex multiplication

The elliptic curves in class 400752.bi do not have complex multiplication.

Modular form 400752.2.a.bi

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + 2 q^{7} + 2 q^{13} + 2 q^{17} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 400752.bi

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400752.bi1 400752bi2 \([0, 0, 0, -42471, 646866]\) \(949104/529\) \(4722185472314112\) \([2]\) \(1474560\) \(1.6979\) \(\Gamma_0(N)\)-optimal*
400752.bi2 400752bi1 \([0, 0, 0, -26136, -1617165]\) \(3538944/23\) \(12832025739984\) \([2]\) \(737280\) \(1.3513\) \(\Gamma_0(N)\)-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 400752.bi1.