# Properties

 Label 76664.b Number of curves $4$ Conductor $76664$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 76664.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76664.b1 76664b4 $$[0, 0, 0, -409331, 100799470]$$ $$1443468546/7$$ $$36782253799424$$ $$$$ $$414720$$ $$1.8034$$
76664.b2 76664b3 $$[0, 0, 0, -80771, -6990114]$$ $$11090466/2401$$ $$12616313053202432$$ $$$$ $$414720$$ $$1.8034$$
76664.b3 76664b2 $$[0, 0, 0, -26011, 1519590]$$ $$740772/49$$ $$128737888297984$$ $$[2, 2]$$ $$207360$$ $$1.4569$$
76664.b4 76664b1 $$[0, 0, 0, 1369, 101306]$$ $$432/7$$ $$-4597781724928$$ $$$$ $$103680$$ $$1.1103$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 76664.b 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 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. 