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Properties

Label 35904.d
Number of curves $4$
Conductor $35904$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

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.c
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 35904.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + q^{11} - 6 q^{13} + 2 q^{15} - q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 35904.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35904.d1 35904ca4 \([0, -1, 0, -30972129, -66332699775]\) \(12534210458299016895673/315581882565708\) \(82727897023304957952\) \([2]\) \(2949120\) \(2.9300\)  
35904.d2 35904ca2 \([0, -1, 0, -2009569, -952616831]\) \(3423676911662954233/483711578981136\) \(126802088160430915584\) \([2, 2]\) \(1474560\) \(2.5834\)  
35904.d3 35904ca1 \([0, -1, 0, -529889, 133764225]\) \(62768149033310713/6915442583808\) \(1812841780689764352\) \([2]\) \(737280\) \(2.2368\) \(\Gamma_0(N)\)-optimal
35904.d4 35904ca3 \([0, -1, 0, 3278111, -5124596351]\) \(14861225463775641287/51859390496937804\) \(-13594628062429263691776\) \([2]\) \(2949120\) \(2.9300\)