Properties

Label 33600ey
Number of curves $8$
Conductor $33600$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ey1")
 
E.isogeny_class()
 

Elliptic curves in class 33600ey

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.dg7 33600ey1 \([0, -1, 0, -796033, -272996063]\) \(13619385906841/6048000\) \(24772608000000000\) \([2]\) \(442368\) \(2.1043\) \(\Gamma_0(N)\)-optimal
33600.dg6 33600ey2 \([0, -1, 0, -924033, -179172063]\) \(21302308926361/8930250000\) \(36578304000000000000\) \([2, 2]\) \(884736\) \(2.4508\)  
33600.dg5 33600ey3 \([0, -1, 0, -2356033, 1058643937]\) \(353108405631241/86318776320\) \(353561707806720000000\) \([2]\) \(1327104\) \(2.6536\)  
33600.dg8 33600ey4 \([0, -1, 0, 3075967, -1319172063]\) \(785793873833639/637994920500\) \(-2613227194368000000000\) \([2]\) \(1769472\) \(2.7974\)  
33600.dg4 33600ey5 \([0, -1, 0, -6972033, 6963515937]\) \(9150443179640281/184570312500\) \(756000000000000000000\) \([2]\) \(1769472\) \(2.7974\)  
33600.dg2 33600ey6 \([0, -1, 0, -35124033, 80127827937]\) \(1169975873419524361/108425318400\) \(444110104166400000000\) \([2, 2]\) \(2654208\) \(3.0001\)  
33600.dg3 33600ey7 \([0, -1, 0, -32564033, 92300627937]\) \(-932348627918877961/358766164249920\) \(-1469506208767672320000000\) \([2]\) \(5308416\) \(3.3467\)  
33600.dg1 33600ey8 \([0, -1, 0, -561972033, 5127858515937]\) \(4791901410190533590281/41160000\) \(168591360000000000\) \([2]\) \(5308416\) \(3.3467\)  

Rank

sage: E.rank()
 

The elliptic curves in class 33600ey have rank \(1\).

Complex multiplication

The elliptic curves in class 33600ey do not have complex multiplication.

Modular form 33600.2.a.ey

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} + 2 q^{13} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with Cremona labels.