Properties

Label 12138ba
Number of curves $4$
Conductor $12138$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ba1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12138ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.bb4 12138ba1 \([1, 0, 0, 283, 495105]\) \(103823/4386816\) \(-105887073890304\) \([2]\) \(55296\) \(1.3700\) \(\Gamma_0(N)\)-optimal
12138.bb3 12138ba2 \([1, 0, 0, -92197, 10575425]\) \(3590714269297/73410624\) \(1771954002133056\) \([2, 2]\) \(110592\) \(1.7166\)  
12138.bb2 12138ba3 \([1, 0, 0, -196237, -17702647]\) \(34623662831857/14438442312\) \(348508897558419528\) \([2]\) \(221184\) \(2.0632\)  
12138.bb1 12138ba4 \([1, 0, 0, -1467837, 684363897]\) \(14489843500598257/6246072\) \(150764993878968\) \([2]\) \(221184\) \(2.0632\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12138.2.a.ba

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.