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Properties

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

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 12138.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.b1 12138d2 \([1, 1, 0, -864, -14094]\) \(-14544652121/8168202\) \(-40130376426\) \([]\) \(16000\) \(0.73786\)  
12138.b2 12138d1 \([1, 1, 0, -14, 84]\) \(-68921/672\) \(-3301536\) \([]\) \(3200\) \(-0.066863\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12138.2.a.b

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 LMFDB 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 LMFDB labels.