# Properties

 Label 112.b Number of curves $4$ Conductor $112$ CM no Rank $0$ Graph # Learn more

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

sage: E.isogeny_class()

## Elliptic curves in class 112.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112.b1 112b3 $$[0, 0, 0, -299, -1990]$$ $$1443468546/7$$ $$14336$$ $$$$ $$16$$ $$-0.0020328$$
112.b2 112b4 $$[0, 0, 0, -59, 138]$$ $$11090466/2401$$ $$4917248$$ $$$$ $$16$$ $$-0.0020328$$
112.b3 112b2 $$[0, 0, 0, -19, -30]$$ $$740772/49$$ $$50176$$ $$[2, 2]$$ $$8$$ $$-0.34861$$
112.b4 112b1 $$[0, 0, 0, 1, -2]$$ $$432/7$$ $$-1792$$ $$$$ $$4$$ $$-0.69518$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 112.b 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 