Properties

Label 12138.ba
Number of curves $6$
Conductor $12138$
CM no
Rank $1$
Graph

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

Elliptic curves in class 12138.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.ba1 12138w5 \([1, 0, 0, -3964676562, -96086246480388]\) \(285531136548675601769470657/17941034271597192\) \(433052952662041962106248\) \([2]\) \(8847360\) \(3.9957\)  
12138.ba2 12138w3 \([1, 0, 0, -248263722, -1495363594140]\) \(70108386184777836280897/552468975892674624\) \(13335258025968770331349056\) \([2, 2]\) \(4423680\) \(3.6492\)  
12138.ba3 12138w6 \([1, 0, 0, -84562562, -3437874298932]\) \(-2770540998624539614657/209924951154647363208\) \(-5067077993316930400101161352\) \([2]\) \(8847360\) \(3.9957\)  
12138.ba4 12138w2 \([1, 0, 0, -26219242, 12984558500]\) \(82582985847542515777/44772582831427584\) \(1080701307401798677303296\) \([2, 2]\) \(2211840\) \(3.3026\)  
12138.ba5 12138w1 \([1, 0, 0, -20300522, 35159634852]\) \(38331145780597164097/55468445663232\) \(1338873434519013163008\) \([4]\) \(1105920\) \(2.9560\) \(\Gamma_0(N)\)-optimal
12138.ba6 12138w4 \([1, 0, 0, 101125718, 102151499492]\) \(4738217997934888496063/2928751705237796928\) \(-70692946369044984757588032\) \([2]\) \(4423680\) \(3.6492\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12138.2.a.ba

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.