# Properties

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

# Related objects

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)

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

## 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.