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Properties

Label 93138q
Number of curves $4$
Conductor $93138$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -8672, 196910]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -8672, 196910]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -8672, 196910]) 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 93138q have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(19\)\(1\)
\(43\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 7 T + 17 T^{2}\) 1.17.h
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 7 T + 29 T^{2}\) 1.29.ah
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 93138q do not have complex multiplication.

Modular form 93138.2.a.q

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);
 
\(q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} - 6 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + O(q^{20})\) 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 93138q

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
93138.l4 93138q1 \([1, 0, 1, -8672, 196910]\) \(1532808577/528384\) \(24858290786304\) \([2]\) \(414720\) \(1.2721\) \(\Gamma_0(N)\)-optimal
93138.l2 93138q2 \([1, 0, 1, -124192, 16831790]\) \(4502751117697/1065024\) \(50104992366144\) \([2, 2]\) \(829440\) \(1.6187\)  
93138.l3 93138q3 \([1, 0, 1, -109752, 20898094]\) \(-3107661785857/2215383048\) \(-104224647245625288\) \([2]\) \(1658880\) \(1.9653\)  
93138.l1 93138q4 \([1, 0, 1, -1986952, 1077859886]\) \(18440127492397057/1032\) \(48551349192\) \([2]\) \(1658880\) \(1.9653\)