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Properties

Label 447330.cv
Number of curves $8$
Conductor $447330$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cv1")
 
E.isogeny_class()
 

Elliptic curves in class 447330.cv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
447330.cv1 447330cv8 \([1, 0, 0, -284296704640, -58345320317497720]\) \(2541202004362280513777612137989837711361/53305632120\) \(53305632120\) \([2]\) \(880803840\) \(4.5471\)  
447330.cv2 447330cv6 \([1, 0, 0, -17768544040, -911646740492800]\) \(620410645598305764158303085551928961/2841490415712775694400\) \(2841490415712775694400\) \([2, 2]\) \(440401920\) \(4.2005\)  
447330.cv3 447330cv7 \([1, 0, 0, -17768255440, -911677835353480]\) \(-620380415613717203478802928936674561/41986862587247533428179923320\) \(-41986862587247533428179923320\) \([2]\) \(880803840\) \(4.5471\)  
447330.cv4 447330cv4 \([1, 0, 0, -1110552040, -14244063870400]\) \(151474823388097505397360613560961/10250665575123920693760000\) \(10250665575123920693760000\) \([2, 4]\) \(220200960\) \(3.8540\)  
447330.cv5 447330cv5 \([1, 0, 0, -1041360040, -16096181488000]\) \(-124889646897589160042815246392961/39650301200798232859676174400\) \(-39650301200798232859676174400\) \([4]\) \(440401920\) \(4.2005\)  
447330.cv6 447330cv3 \([1, 0, 0, -379129320, 2672945013312]\) \(6026786794227239560650117386881/401168535519375000000000000\) \(401168535519375000000000000\) \([4]\) \(220200960\) \(3.8540\) \(\Gamma_0(N)\)-optimal*
447330.cv7 447330cv2 \([1, 0, 0, -73752040, -193142910400]\) \(44365545422241845117106360961/9560123635669401600000000\) \(9560123635669401600000000\) \([2, 4]\) \(110100480\) \(3.5074\) \(\Gamma_0(N)\)-optimal*
447330.cv8 447330cv1 \([1, 0, 0, 10134040, -18374651328]\) \(115099000398621243971890559/212476839747712450560000\) \(-212476839747712450560000\) \([4]\) \(55050240\) \(3.1608\) \(\Gamma_0(N)\)-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 447330.cv1.

Rank

sage: E.rank()
 

The elliptic curves in class 447330.cv have rank \(0\).

Complex multiplication

The elliptic curves in class 447330.cv do not have complex multiplication.

Modular form 447330.2.a.cv

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} + q^{13} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 8 & 16 \\ 2 & 1 & 2 & 2 & 4 & 8 & 4 & 8 \\ 4 & 2 & 1 & 4 & 8 & 16 & 8 & 16 \\ 4 & 2 & 4 & 1 & 2 & 4 & 2 & 4 \\ 8 & 4 & 8 & 2 & 1 & 8 & 4 & 8 \\ 16 & 8 & 16 & 4 & 8 & 1 & 2 & 4 \\ 8 & 4 & 8 & 2 & 4 & 2 & 1 & 2 \\ 16 & 8 & 16 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.