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Properties

Label 12138e
Number of curves $2$
Conductor $12138$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 12138e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.e2 12138e1 \([1, 1, 0, 15456, -3600720]\) \(3442951/49392\) \(-5857292395939824\) \([2]\) \(130560\) \(1.7062\) \(\Gamma_0(N)\)-optimal
12138.e1 12138e2 \([1, 1, 0, -279324, -53418540]\) \(20324066489/1411788\) \(167420940983946636\) \([2]\) \(261120\) \(2.0528\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12138e have rank \(0\).

Complex multiplication

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

Modular form 12138.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{10} - 6 q^{11} - q^{12} + 6 q^{13} - q^{14} - 2 q^{15} + q^{16} - q^{18} - 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.