# Properties

 Label 91.b Number of curves $3$ Conductor $91$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 91.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
91.b1 91b3 $$[0, 1, 1, -117, -1245]$$ $$-178643795968/524596891$$ $$-524596891$$ $$[]$$ $$36$$ $$0.36010$$
91.b2 91b1 $$[0, 1, 1, -7, 5]$$ $$-43614208/91$$ $$-91$$ $$$$ $$4$$ $$-0.73851$$ $$\Gamma_0(N)$$-optimal
91.b3 91b2 $$[0, 1, 1, 13, 42]$$ $$224755712/753571$$ $$-753571$$ $$$$ $$12$$ $$-0.18920$$

## Rank

sage: E.rank()

The elliptic curves in class 91.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 91.b do not have complex multiplication.

## Modular form91.2.a.b

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} + 4q^{12} + q^{13} + 6q^{15} + 4q^{16} - 6q^{17} - 7q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 