Properties

Label 33600.dz
Number of curves $6$
Conductor $33600$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("dz1")
 
E.isogeny_class()
 

Elliptic curves in class 33600.dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.dz1 33600gi4 \([0, 1, 0, -2150433, -1214488737]\) \(268498407453697/252\) \(1032192000000\) \([2]\) \(393216\) \(2.0335\)  
33600.dz2 33600gi6 \([0, 1, 0, -1462433, 673687263]\) \(84448510979617/933897762\) \(3825245233152000000\) \([2]\) \(786432\) \(2.3801\)  
33600.dz3 33600gi3 \([0, 1, 0, -166433, -9304737]\) \(124475734657/63011844\) \(258096513024000000\) \([2, 2]\) \(393216\) \(2.0335\)  
33600.dz4 33600gi2 \([0, 1, 0, -134433, -19000737]\) \(65597103937/63504\) \(260112384000000\) \([2, 2]\) \(196608\) \(1.6870\)  
33600.dz5 33600gi1 \([0, 1, 0, -6433, -440737]\) \(-7189057/16128\) \(-66060288000000\) \([2]\) \(98304\) \(1.3404\) \(\Gamma_0(N)\)-optimal
33600.dz6 33600gi5 \([0, 1, 0, 617567, -71240737]\) \(6359387729183/4218578658\) \(-17279298183168000000\) \([2]\) \(786432\) \(2.3801\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 33600.2.a.dz

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.