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Properties

Label 38b
Number of curves $2$
Conductor $38$
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, 1, 1, 0, 1]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, 0, 1]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, 0, 1]) 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 38b have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(5\) \( 1 + 4 T + 5 T^{2}\) 1.5.e
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 38b do not have complex multiplication.

Modular form 38.2.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);
 
\(q + q^{2} - q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} - 4 q^{10} + 2 q^{11} - q^{12} - q^{13} + 3 q^{14} + 4 q^{15} + q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

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 38b

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
38.b2 38b1 \([1, 1, 1, 0, 1]\) \(-1/608\) \(-608\) \([5]\) \(2\) \(-0.78693\) \(\Gamma_0(N)\)-optimal
38.b1 38b2 \([1, 1, 1, -70, -279]\) \(-37966934881/4952198\) \(-4952198\) \([]\) \(10\) \(0.017785\)