# Properties

 Label 2112g Number of curves 4 Conductor 2112 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.m1")

sage: E.isogeny_class()

## Elliptic curves in class 2112g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.m4 2112g1 [0, -1, 0, 3, 45]  384 $$\Gamma_0(N)$$-optimal
2112.m3 2112g2 [0, -1, 0, -177, 945] [2, 2] 768
2112.m2 2112g3 [0, -1, 0, -417, -1887]  1536
2112.m1 2112g4 [0, -1, 0, -2817, 58497]  1536

## Rank

sage: E.rank()

The elliptic curves in class 2112g have rank $$0$$.

## Modular form2112.2.a.m

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + 4q^{7} + q^{9} + q^{11} - 6q^{13} - 2q^{15} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 