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Properties

Label 436800.mr
Number of curves $8$
Conductor $436800$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("mr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 436800.mr have rank \(1\).

Complex multiplication

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

Modular form 436800.2.a.mr

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 436800.mr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.mr1 436800mr8 \([0, 1, 0, -64304728033, -6276448294455937]\) \(7179471593960193209684686321/49441793310\) \(202513585397760000000\) \([2]\) \(509607936\) \(4.4322\)  
436800.mr2 436800mr6 \([0, 1, 0, -4019048033, -98070379655937]\) \(1752803993935029634719121/4599740941532100\) \(18840538896515481600000000\) \([2, 2]\) \(254803968\) \(4.0856\)  
436800.mr3 436800mr7 \([0, 1, 0, -3969656033, -100598212823937]\) \(-1688971789881664420008241/89901485966373558750\) \(-368236486518266096640000000000\) \([2]\) \(509607936\) \(4.4322\)  
436800.mr4 436800mr5 \([0, 1, 0, -794248033, -8601647655937]\) \(13527956825588849127121/25701087819771000\) \(105271655709782016000000000\) \([2]\) \(169869312\) \(3.8829\)  
436800.mr5 436800mr3 \([0, 1, 0, -254280033, -1492786151937]\) \(443915739051786565201/21894701746029840\) \(89680698351738224640000000\) \([2]\) \(127401984\) \(3.7391\)  
436800.mr6 436800mr2 \([0, 1, 0, -66248033, -36727655937]\) \(7850236389974007121/4400862921000000\) \(18025934524416000000000000\) \([2, 2]\) \(84934656\) \(3.5363\)  
436800.mr7 436800mr1 \([0, 1, 0, -41160033, 101080728063]\) \(1882742462388824401/11650189824000\) \(47719177519104000000000\) \([2]\) \(42467328\) \(3.1898\) \(\Gamma_0(N)\)-optimal*
436800.mr8 436800mr4 \([0, 1, 0, 260343967, -291142823937]\) \(476437916651992691759/284661685546875000\) \(-1165974264000000000000000000\) \([2]\) \(169869312\) \(3.8829\)  
*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.mr1.