# Properties

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

# Related objects

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)

$$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} + 2q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.