Properties

Label 12138a
Number of curves $4$
Conductor $12138$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 12138a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.d3 12138a1 \([1, 1, 0, -156499, 24133165]\) \(-3574558889/64512\) \(-7650341088574464\) \([2]\) \(108800\) \(1.8441\) \(\Gamma_0(N)\)-optimal
12138.d2 12138a2 \([1, 1, 0, -2514739, 1533878413]\) \(14830727012009/4704\) \(557837371041888\) \([2]\) \(217600\) \(2.1907\)  
12138.d4 12138a3 \([1, 1, 0, 924361, -985881375]\) \(736558976791/3969746172\) \(-470763768769574519484\) \([2]\) \(544000\) \(2.6488\)  
12138.d1 12138a4 \([1, 1, 0, -11014229, -12664210113]\) \(1246079601667529/137282971014\) \(16280096011749462857958\) \([2]\) \(1088000\) \(2.9954\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12138a have rank \(1\).

Complex multiplication

The elliptic curves in class 12138a do not have complex multiplication.

Modular form 12138.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{10} - q^{12} + 4 q^{13} + q^{14} - 2 q^{15} + q^{16} - q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.