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Properties

Label 12138n
Number of curves $2$
Conductor $12138$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 12138n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.o2 12138n1 \([1, 0, 1, -4197, 441712]\) \(-68921/672\) \(-79691053005984\) \([]\) \(54400\) \(1.3497\) \(\Gamma_0(N)\)-optimal
12138.o1 12138n2 \([1, 0, 1, -249847, -67495252]\) \(-14544652121/8168202\) \(-968649729978548394\) \([]\) \(272000\) \(2.1545\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12138.2.a.n

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.