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Properties

Label 400710t
Number of curves $2$
Conductor $400710$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("t1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 400710t have rank \(0\).

Complex multiplication

The elliptic curves in class 400710t do not have complex multiplication.

Modular form 400710.2.a.t

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

Elliptic curves in class 400710t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400710.t2 400710t1 \([1, 0, 1, 54162627, -343089255224]\) \(373509178976018769839/1297056833088603120\) \(-61021181419723284839748720\) \([2]\) \(193536000\) \(3.6308\) \(\Gamma_0(N)\)-optimal*
400710.t1 400710t2 \([1, 0, 1, -530072553, -4081960713152]\) \(350112854843907984798481/49611975051499911900\) \(2334039074447833726757883900\) \([2]\) \(387072000\) \(3.9773\) \(\Gamma_0(N)\)-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 400710t1.