# Properties

 Label 112a Number of curves $2$ Conductor $112$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 112a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112.a2 112a1 $$[0, 1, 0, 0, 4]$$ $$-4/7$$ $$-7168$$ $$$$ $$8$$ $$-0.58124$$ $$\Gamma_0(N)$$-optimal
112.a1 112a2 $$[0, 1, 0, -40, 84]$$ $$3543122/49$$ $$100352$$ $$$$ $$16$$ $$-0.23467$$

## Rank

sage: E.rank()

The elliptic curves in class 112a have rank $$1$$.

## Complex multiplication

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

## Modular form112.2.a.a

sage: E.q_eigenform(10)

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

## 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. 