# Properties

 Label 112b Number of curves $4$ Conductor $112$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 112b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112.b4 112b1 [0, 0, 0, 1, -2]  4 $$\Gamma_0(N)$$-optimal
112.b3 112b2 [0, 0, 0, -19, -30] [2, 2] 8
112.b1 112b3 [0, 0, 0, -299, -1990]  16
112.b2 112b4 [0, 0, 0, -59, 138]  16

## Rank

sage: E.rank()

The elliptic curves in class 112b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 112b do not have complex multiplication.

## Modular form112.2.a.b

sage: E.q_eigenform(10)

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

## 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. 