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Properties

Label 35904.cz
Number of curves $6$
Conductor $35904$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 35904.cz have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(11\)\(1 - T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 35904.cz do not have complex multiplication.

Modular form 35904.2.a.cz

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 35904.cz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35904.cz1 35904bm6 \([0, 1, 0, -25276417, -48921097633]\) \(6812873765474836663297/74052\) \(19412287488\) \([2]\) \(786432\) \(2.4776\)  
35904.cz2 35904bm4 \([0, 1, 0, -1579777, -764785825]\) \(1663303207415737537/5483698704\) \(1437518713061376\) \([2, 2]\) \(393216\) \(2.1311\)  
35904.cz3 35904bm5 \([0, 1, 0, -1558017, -786854817]\) \(-1595514095015181697/95635786040388\) \(-25070347495771471872\) \([2]\) \(786432\) \(2.4776\)  
35904.cz4 35904bm2 \([0, 1, 0, -100097, -11628705]\) \(423108074414017/23284318464\) \(6103844379426816\) \([2, 2]\) \(196608\) \(1.7845\)  
35904.cz5 35904bm1 \([0, 1, 0, -18177, 708447]\) \(2533811507137/625016832\) \(163844412407808\) \([2]\) \(98304\) \(1.4379\) \(\Gamma_0(N)\)-optimal
35904.cz6 35904bm3 \([0, 1, 0, 68863, -46738593]\) \(137763859017023/3683199928848\) \(-965528762147930112\) \([2]\) \(393216\) \(2.1311\)