Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -5, -7]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -5, -7]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -5, -7]) 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 curve 174.d1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(29\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 7 T + 17 T^{2}\) 1.17.h
\(19\) \( 1 + 3 T + 19 T^{2}\) 1.19.d
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 174.d do not have complex multiplication.

Modular form 174.2.a.d

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);
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + 6 q^{11} - q^{12} - 4 q^{13} + q^{14} - q^{15} + q^{16} - 7 q^{17} + q^{18} - 3 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

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

Elliptic curves in class 174.d

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
174.d1 174c1 \([1, 1, 1, -5, -7]\) \(-13997521/1566\) \(-1566\) \([]\) \(12\) \(-0.64975\) \(\Gamma_0(N)\)-optimal