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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 1, -3283, -74657]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 1, -3283, -74657]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 1, -3283, -74657]) 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 175.b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T^{2}\) 1.2.a
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 175.b do not have complex multiplication.

Modular form 175.2.a.b

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);
 
\(q - q^{3} - 2 q^{4} - q^{7} - 2 q^{9} - 3 q^{11} + 2 q^{12} - 5 q^{13} + 4 q^{16} - 3 q^{17} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 175.b

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
175.b1 175b3 \([0, -1, 1, -3283, -74657]\) \(-250523582464/13671875\) \(-213623046875\) \([]\) \(144\) \(0.93218\)  
175.b2 175b1 \([0, -1, 1, -33, 93]\) \(-262144/35\) \(-546875\) \([]\) \(16\) \(-0.16643\) \(\Gamma_0(N)\)-optimal
175.b3 175b2 \([0, -1, 1, 217, -282]\) \(71991296/42875\) \(-669921875\) \([]\) \(48\) \(0.38287\)