# Properties

 Label 5491.b Number of curves $3$ Conductor $5491$ CM no Rank $0$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 5491.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5491.b1 5491a3 $$[0, -1, 1, -222337, -40278040]$$ $$-50357871050752/19$$ $$-458613811$$ $$[]$$ $$15120$$ $$1.4500$$
5491.b2 5491a2 $$[0, -1, 1, -2697, -56465]$$ $$-89915392/6859$$ $$-165559585771$$ $$[]$$ $$5040$$ $$0.90074$$
5491.b3 5491a1 $$[0, -1, 1, 193, -110]$$ $$32768/19$$ $$-458613811$$ $$[]$$ $$1680$$ $$0.35143$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5491.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 5491.b do not have complex multiplication.

## Modular form5491.2.a.b

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{12} - 4q^{13} - 6q^{15} + 4q^{16} + q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 