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Properties

Label 74970bg
Number of curves $4$
Conductor $74970$
CM no
Rank $2$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, 1755, -3515]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, 1755, -3515]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, 1755, -3515]) 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 74970bg have rank \(2\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(19\) \( 1 + 3 T + 19 T^{2}\) 1.19.d
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 74970bg do not have complex multiplication.

Modular form 74970.2.a.bg

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);
 
\(q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4 q^{11} - 2 q^{13} + q^{16} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 74970bg

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
74970.b4 74970bg1 \([1, -1, 0, 1755, -3515]\) \(6967871/4080\) \(-349925773680\) \([2]\) \(98304\) \(0.90449\) \(\Gamma_0(N)\)-optimal
74970.b3 74970bg2 \([1, -1, 0, -7065, -22919]\) \(454756609/260100\) \(22307768072100\) \([2, 2]\) \(196608\) \(1.2511\)  
74970.b2 74970bg3 \([1, -1, 0, -73215, 7610791]\) \(506071034209/2505630\) \(214898165761230\) \([2]\) \(393216\) \(1.5976\)  
74970.b1 74970bg4 \([1, -1, 0, -82035, -9004325]\) \(711882749089/1721250\) \(147624935771250\) \([2]\) \(393216\) \(1.5976\)