# Properties

 Label 2112.j Number of curves 4 Conductor 2112 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.j1")

sage: E.isogeny_class()

## Elliptic curves in class 2112.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.j1 2112v3 [0, -1, 0, -9377, -346047] [2] 3072
2112.j2 2112v2 [0, -1, 0, -737, -2175] [2, 2] 1536
2112.j3 2112v1 [0, -1, 0, -417, 3393] [2] 768 $$\Gamma_0(N)$$-optimal
2112.j4 2112v4 [0, -1, 0, 2783, -19775] [2] 3072

## Rank

sage: E.rank()

The elliptic curves in class 2112.j have rank $$1$$.

## Modular form2112.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} - 4q^{7} + q^{9} + q^{11} + 2q^{13} - 2q^{15} - 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.