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Properties

Label 2535.j
Number of curves $8$
Conductor $2535$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("j1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 2535.j have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 2535.j do not have complex multiplication.

Modular form 2535.2.a.j

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3 q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} + q^{15} - q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 2535.j

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2535.j1 2535a7 \([1, 1, 0, -365043, -85043772]\) \(1114544804970241/405\) \(1954857645\) \([2]\) \(9216\) \(1.5733\)  
2535.j2 2535a5 \([1, 1, 0, -22818, -1335537]\) \(272223782641/164025\) \(791717346225\) \([2, 2]\) \(4608\) \(1.2268\)  
2535.j3 2535a8 \([1, 1, 0, -18593, -1840002]\) \(-147281603041/215233605\) \(-1038891501716445\) \([2]\) \(9216\) \(1.5733\)  
2535.j4 2535a4 \([1, 1, 0, -13523, 599682]\) \(56667352321/15\) \(72402135\) \([2]\) \(2304\) \(0.88020\)  
2535.j5 2535a3 \([1, 1, 0, -1693, -13112]\) \(111284641/50625\) \(244357205625\) \([2, 2]\) \(2304\) \(0.88020\)  
2535.j6 2535a2 \([1, 1, 0, -848, 9027]\) \(13997521/225\) \(1086032025\) \([2, 2]\) \(1152\) \(0.53362\)  
2535.j7 2535a1 \([1, 1, 0, -3, 408]\) \(-1/15\) \(-72402135\) \([2]\) \(576\) \(0.18705\) \(\Gamma_0(N)\)-optimal
2535.j8 2535a6 \([1, 1, 0, 5912, -90683]\) \(4733169839/3515625\) \(-16969250390625\) \([2]\) \(4608\) \(1.2268\)