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Properties

Label 402930dj
Number of curves $6$
Conductor $402930$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dj1")
 
E.isogeny_class()
 

Elliptic curves in class 402930dj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
402930.dj5 402930dj1 \([1, -1, 1, -920228, 1169881287]\) \(-66730743078481/419010969600\) \(-541139245898032742400\) \([2]\) \(17694720\) \(2.6619\) \(\Gamma_0(N)\)-optimal*
402930.dj4 402930dj2 \([1, -1, 1, -23222948, 42991941831]\) \(1072487167529950801/2554882560000\) \(3299548990796720640000\) \([2, 2]\) \(35389440\) \(3.0085\) \(\Gamma_0(N)\)-optimal*
402930.dj1 402930dj3 \([1, -1, 1, -371354468, 2754518724807]\) \(4385367890843575421521/24975000000\) \(32254412525775000000\) \([2]\) \(70778880\) \(3.3551\) \(\Gamma_0(N)\)-optimal*
402930.dj3 402930dj4 \([1, -1, 1, -31934948, 7809401031]\) \(2788936974993502801/1593609593601600\) \(2058095745227573747150400\) \([2, 2]\) \(70778880\) \(3.3551\)  
402930.dj6 402930dj5 \([1, -1, 1, 126841252, 62174371911]\) \(174751791402194852399/102423900876336360\) \(-132277187241819439010052840\) \([2]\) \(141557760\) \(3.7016\)  
402930.dj2 402930dj6 \([1, -1, 1, -330103148, -2298342725049]\) \(3080272010107543650001/15465841417699560\) \(19973638804592531405393640\) \([2]\) \(141557760\) \(3.7016\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 402930dj1.

Rank

sage: E.rank()
 

The elliptic curves in class 402930dj have rank \(1\).

Complex multiplication

The elliptic curves in class 402930dj do not have complex multiplication.

Modular form 402930.2.a.dj

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2 q^{13} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.