# Properties

 Label 12342j Number of curves $2$ Conductor $12342$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 12342j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12342.q2 12342j1 [1, 0, 1, -1455, -17426] [2] 11520 $$\Gamma_0(N)$$-optimal
12342.q1 12342j2 [1, 0, 1, -22025, -1259854] [2] 23040

## Rank

sage: E.rank()

The elliptic curves in class 12342j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 12342j do not have complex multiplication.

## Modular form 12342.2.a.j

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 2q^{7} - q^{8} + q^{9} - 2q^{10} + q^{12} - 4q^{13} - 2q^{14} + 2q^{15} + q^{16} - q^{17} - q^{18} - 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.