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Properties

Label 400710.x
Number of curves $6$
Conductor $400710$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 400710.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400710.x1 400710x4 \([1, 0, 1, -123102813, -525724902344]\) \(4385367890843575421521/24975000000\) \(1174970877975000000\) \([2]\) \(47775744\) \(3.0790\)  
400710.x2 400710x5 \([1, 0, 1, -109428133, 438672429128]\) \(3080272010107543650001/15465841417699560\) \(727604134901964793512360\) \([2]\) \(95551488\) \(3.4256\) \(\Gamma_0(N)\)-optimal*
400710.x3 400710x3 \([1, 0, 1, -10586333, -1489874632]\) \(2788936974993502801/1593609593601600\) \(74972767301039235009600\) \([2, 2]\) \(47775744\) \(3.0790\) \(\Gamma_0(N)\)-optimal*
400710.x4 400710x2 \([1, 0, 1, -7698333, -8205052232]\) \(1072487167529950801/2554882560000\) \(120196700886735360000\) \([2, 2]\) \(23887872\) \(2.7325\) \(\Gamma_0(N)\)-optimal*
400710.x5 400710x1 \([1, 0, 1, -305053, -223267144]\) \(-66730743078481/419010969600\) \(-19712740213496217600\) \([2]\) \(11943936\) \(2.3859\) \(\Gamma_0(N)\)-optimal*
400710.x6 400710x6 \([1, 0, 1, 42047467, -11869259992]\) \(174751791402194852399/102423900876336360\) \(-4818622652183916108533160\) \([2]\) \(95551488\) \(3.4256\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 400710.x1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 400710.2.a.x

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.