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Properties

Label 436800.or
Number of curves $4$
Conductor $436800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("or1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 436800.or have rank \(0\).

Complex multiplication

The elliptic curves in class 436800.or do not have complex multiplication.

Modular form 436800.2.a.or

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} + 4 q^{11} + q^{13} + 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 436800.or

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.or1 436800or4 \([0, 1, 0, -252375301633, -48799923218075137]\) \(434014578033107719741685694649/103121648659575000\) \(422386272909619200000000000\) \([2]\) \(2123366400\) \(4.9276\)  
436800.or2 436800or2 \([0, 1, 0, -15775301633, -762315418075137]\) \(105997782562506306791694649/51649016225625000000\) \(211554370460160000000000000000\) \([2, 2]\) \(1061683200\) \(4.5810\)  
436800.or3 436800or3 \([0, 1, 0, -13147013633, -1024657984795137]\) \(-61354313914516350666047929/75227254486083984375000\) \(-308130834375000000000000000000000\) \([2]\) \(2123366400\) \(4.9276\)  
436800.or4 436800or1 \([0, 1, 0, -1152069633, -7625037787137]\) \(41285728533151645510969/17760741842188800000\) \(72747998585605324800000000000\) \([2]\) \(530841600\) \(4.2344\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 436800.or1.