# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12138.ba1")

sage: E.isogeny_class()

## Elliptic curves in class 12138w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12138.ba5 12138w1 [1, 0, 0, -20300522, 35159634852] [4] 1105920 $$\Gamma_0(N)$$-optimal
12138.ba4 12138w2 [1, 0, 0, -26219242, 12984558500] [2, 2] 2211840
12138.ba2 12138w3 [1, 0, 0, -248263722, -1495363594140] [2, 2] 4423680
12138.ba6 12138w4 [1, 0, 0, 101125718, 102151499492] [2] 4423680
12138.ba1 12138w5 [1, 0, 0, -3964676562, -96086246480388] [2] 8847360
12138.ba3 12138w6 [1, 0, 0, -84562562, -3437874298932] [2] 8847360

## Rank

sage: E.rank()

The elliptic curves in class 12138w have rank $$1$$.

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.