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Properties

Label 35280dx
Number of curves $8$
Conductor $35280$
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([0, 0, 0, 10437, -1260182]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 10437, -1260182]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 10437, -1260182]) 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 35280dx have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 35280dx do not have complex multiplication.

Modular form 35280.2.a.dx

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);
 
\(q - q^{5} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 35280dx

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
35280.bp8 35280dx1 \([0, 0, 0, 10437, -1260182]\) \(357911/2160\) \(-758803748290560\) \([2]\) \(110592\) \(1.5371\) \(\Gamma_0(N)\)-optimal
35280.bp6 35280dx2 \([0, 0, 0, -130683, -16472918]\) \(702595369/72900\) \(25609626504806400\) \([2, 2]\) \(221184\) \(1.8837\)  
35280.bp7 35280dx3 \([0, 0, 0, -95403, 37117402]\) \(-273359449/1536000\) \(-539593776562176000\) \([2]\) \(331776\) \(2.0864\)  
35280.bp5 35280dx4 \([0, 0, 0, -483483, 111452362]\) \(35578826569/5314410\) \(1866941772200386560\) \([2]\) \(442368\) \(2.2303\)  
35280.bp4 35280dx5 \([0, 0, 0, -2035803, -1118013302]\) \(2656166199049/33750\) \(11856308567040000\) \([2]\) \(442368\) \(2.2303\)  
35280.bp3 35280dx6 \([0, 0, 0, -2353323, 1386901978]\) \(4102915888729/9000000\) \(3161682284544000000\) \([2, 2]\) \(663552\) \(2.4330\)  
35280.bp1 35280dx7 \([0, 0, 0, -37633323, 88860133978]\) \(16778985534208729/81000\) \(28455140560896000\) \([2]\) \(1327104\) \(2.7796\)  
35280.bp2 35280dx8 \([0, 0, 0, -3200043, 299882842]\) \(10316097499609/5859375000\) \(2058386904000000000000\) \([2]\) \(1327104\) \(2.7796\)