# Properties

 Label 56.b Number of curves $2$ Conductor $56$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 56.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
56.b1 56b2 $$[0, -1, 0, -40, -84]$$ $$3543122/49$$ $$100352$$ $$$$ $$8$$ $$-0.23467$$
56.b2 56b1 $$[0, -1, 0, 0, -4]$$ $$-4/7$$ $$-7168$$ $$$$ $$4$$ $$-0.58124$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 56.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 56.b do not have complex multiplication.

## Modular form56.2.a.b

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 