# Properties

 Label 76664b Number of curves $4$ Conductor $76664$ CM no Rank $1$ Graph

# Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 76664b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
76664.b4 76664b1 [0, 0, 0, 1369, 101306] [2] 103680 $$\Gamma_0(N)$$-optimal
76664.b3 76664b2 [0, 0, 0, -26011, 1519590] [2, 2] 207360
76664.b2 76664b3 [0, 0, 0, -80771, -6990114] [2] 414720
76664.b1 76664b4 [0, 0, 0, -409331, 100799470] [2] 414720

## Rank

sage: E.rank()

The elliptic curves in class 76664b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 76664b do not have complex multiplication.

## Modular form 76664.2.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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.