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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ex1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 - 7 T + 29 T^{2}\) 1.29.ah
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 33600.2.a.ex

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 33600ex

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.df7 33600ex1 \([0, -1, 0, -65633, 2299137]\) \(7633736209/3870720\) \(15854469120000000\) \([2]\) \(221184\) \(1.8010\) \(\Gamma_0(N)\)-optimal
33600.df5 33600ex2 \([0, -1, 0, -577633, -167172863]\) \(5203798902289/57153600\) \(234101145600000000\) \([2, 2]\) \(442368\) \(2.1476\)  
33600.df4 33600ex3 \([0, -1, 0, -4289633, 3421051137]\) \(2131200347946769/2058000\) \(8429568000000000\) \([2]\) \(663552\) \(2.3503\)  
33600.df6 33600ex4 \([0, -1, 0, -129633, -420292863]\) \(-58818484369/18600435000\) \(-76187381760000000000\) \([2]\) \(884736\) \(2.4942\)  
33600.df2 33600ex5 \([0, -1, 0, -9217633, -10768452863]\) \(21145699168383889/2593080\) \(10621255680000000\) \([2]\) \(884736\) \(2.4942\)  
33600.df3 33600ex6 \([0, -1, 0, -4321633, 3367451137]\) \(2179252305146449/66177562500\) \(271063296000000000000\) \([2, 2]\) \(1327104\) \(2.6969\)  
33600.df8 33600ex7 \([0, -1, 0, 1166367, 11330539137]\) \(42841933504271/13565917968750\) \(-55566000000000000000000\) \([2]\) \(2654208\) \(3.0435\)  
33600.df1 33600ex8 \([0, -1, 0, -10321633, -8026548863]\) \(29689921233686449/10380965400750\) \(42520434281472000000000\) \([2]\) \(2654208\) \(3.0435\)