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Properties

Label 493680cf
Number of curves $8$
Conductor $493680$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 493680cf have rank \(1\).

Complex multiplication

The elliptic curves in class 493680cf do not have complex multiplication.

Modular form 493680.2.a.cf

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - 4 q^{7} + q^{9} - 2 q^{13} - q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 493680cf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
493680.cf8 493680cf1 \([0, -1, 0, 83969120, -125343545600]\) \(9023321954633914439/6156756739584000\) \(-44675359237464762875904000\) \([2]\) \(159252480\) \(3.6104\) \(\Gamma_0(N)\)-optimal*
493680.cf7 493680cf2 \([0, -1, 0, -367660960, -1045584996608]\) \(757443433548897303481/373234243041000000\) \(2708304809312079876096000000\) \([2, 2]\) \(318504960\) \(3.9570\) \(\Gamma_0(N)\)-optimal*
493680.cf6 493680cf3 \([0, -1, 0, -1511633680, -23172506384960]\) \(-52643812360427830814761/1504091705903677440\) \(-10914161486243531609226608640\) \([2]\) \(477757440\) \(4.1597\)  
493680.cf5 493680cf4 \([0, -1, 0, -3150622240, 67341792313600]\) \(476646772170172569823801/5862293314453125000\) \(42538640205602376000000000000\) \([4]\) \(637009920\) \(4.3036\) \(\Gamma_0(N)\)-optimal*
493680.cf4 493680cf5 \([0, -1, 0, -4810780960, -128335641252608]\) \(1696892787277117093383481/1440538624914939000\) \(10453000384073458687094784000\) \([2]\) \(637009920\) \(4.3036\)  
493680.cf3 493680cf6 \([0, -1, 0, -24349618960, -1462458879507008]\) \(220031146443748723000125481/172266701724057600\) \(1250021254647698251422105600\) \([2, 2]\) \(955514880\) \(4.5063\)  
493680.cf2 493680cf7 \([0, -1, 0, -24513172240, -1441816231246400]\) \(224494757451893010998773801/6152490825146276160000\) \(44644404423421796926610472960000\) \([4]\) \(1911029760\) \(4.8529\)  
493680.cf1 493680cf8 \([0, -1, 0, -389593830160, -93597794815343168]\) \(901247067798311192691198986281/552431869440\) \(4008619028713454960640\) \([2]\) \(1911029760\) \(4.8529\)  
*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 493680cf1.