Properties

Label 35904.bg
Number of curves $6$
Conductor $35904$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bg1")
 
E.isogeny_class()
 

Elliptic curves in class 35904.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35904.bg1 35904bt6 \([0, -1, 0, -25276417, 48921097633]\) \(6812873765474836663297/74052\) \(19412287488\) \([2]\) \(786432\) \(2.4776\)  
35904.bg2 35904bt4 \([0, -1, 0, -1579777, 764785825]\) \(1663303207415737537/5483698704\) \(1437518713061376\) \([2, 2]\) \(393216\) \(2.1311\)  
35904.bg3 35904bt5 \([0, -1, 0, -1558017, 786854817]\) \(-1595514095015181697/95635786040388\) \(-25070347495771471872\) \([2]\) \(786432\) \(2.4776\)  
35904.bg4 35904bt2 \([0, -1, 0, -100097, 11628705]\) \(423108074414017/23284318464\) \(6103844379426816\) \([2, 2]\) \(196608\) \(1.7845\)  
35904.bg5 35904bt1 \([0, -1, 0, -18177, -708447]\) \(2533811507137/625016832\) \(163844412407808\) \([2]\) \(98304\) \(1.4379\) \(\Gamma_0(N)\)-optimal
35904.bg6 35904bt3 \([0, -1, 0, 68863, 46738593]\) \(137763859017023/3683199928848\) \(-965528762147930112\) \([2]\) \(393216\) \(2.1311\)  

Rank

sage: E.rank()
 

The elliptic curves in class 35904.bg have rank \(1\).

Complex multiplication

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

Modular form 35904.2.a.bg

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

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

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

The vertices are labelled with LMFDB labels.