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Properties

Label 7935.j
Number of curves $4$
Conductor $7935$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -15940633, -24488418157]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -15940633, -24488418157]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -15940633, -24488418157]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 7935.j have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(23\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7935.j do not have complex multiplication.

Modular form 7935.2.a.j

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + 6 q^{13} - 4 q^{14} + q^{15} - q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 7935.j

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7935.j1 7935j3 \([1, 0, 1, -15940633, -24488418157]\) \(3026030815665395929/1364501953125\) \(201995259673095703125\) \([2]\) \(633600\) \(2.8540\)  
7935.j2 7935j4 \([1, 0, 1, -8762103, 9808700131]\) \(502552788401502649/10024505152875\) \(1483986532090931530875\) \([4]\) \(633600\) \(2.8540\)  
7935.j3 7935j2 \([1, 0, 1, -1157728, -250367119]\) \(1159246431432649/488076890625\) \(72252896404027640625\) \([2, 2]\) \(316800\) \(2.5074\)  
7935.j4 7935j1 \([1, 0, 1, 241477, -28733047]\) \(10519294081031/8500170375\) \(-1258330278114588375\) \([2]\) \(158400\) \(2.1609\) \(\Gamma_0(N)\)-optimal