Properties

Label 58800.cr
Number of curves $8$
Conductor $58800$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cr1")
 
E.isogeny_class()
 

Elliptic curves in class 58800.cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58800.cr1 58800fe8 \([0, -1, 0, -6884157408, -219846607634688]\) \(4791901410190533590281/41160000\) \(309915701760000000000\) \([2]\) \(31850496\) \(3.9731\)  
58800.cr2 58800fe6 \([0, -1, 0, -430269408, -3434835218688]\) \(1169975873419524361/108425318400\) \(816392338204262400000000\) \([2, 2]\) \(15925248\) \(3.6265\)  
58800.cr3 58800fe7 \([0, -1, 0, -398909408, -3956791058688]\) \(-932348627918877961/358766164249920\) \(-2701342749301685637120000000\) \([2]\) \(31850496\) \(3.9731\)  
58800.cr4 58800fe5 \([0, -1, 0, -85407408, -298432634688]\) \(9150443179640281/184570312500\) \(1389728812500000000000000\) \([2]\) \(10616832\) \(3.4238\)  
58800.cr5 58800fe3 \([0, -1, 0, -28861408, -45346066688]\) \(353108405631241/86318776320\) \(649940333777387520000000\) \([2]\) \(7962624\) \(3.2799\)  
58800.cr6 58800fe2 \([0, -1, 0, -11319408, 7698981312]\) \(21302308926361/8930250000\) \(67240638864000000000000\) \([2, 2]\) \(5308416\) \(3.0772\)  
58800.cr7 58800fe1 \([0, -1, 0, -9751408, 11719333312]\) \(13619385906841/6048000\) \(45538633728000000000\) \([2]\) \(2654208\) \(2.7306\) \(\Gamma_0(N)\)-optimal
58800.cr8 58800fe4 \([0, -1, 0, 37680592, 56502981312]\) \(785793873833639/637994920500\) \(-4803805721721888000000000\) \([2]\) \(10616832\) \(3.4238\)  

Rank

sage: E.rank()
 

The elliptic curves in class 58800.cr have rank \(1\).

Complex multiplication

The elliptic curves in class 58800.cr do not have complex multiplication.

Modular form 58800.2.a.cr

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 2 q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.