# Properties

 Label 77b Number of curves $3$ Conductor $77$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 77b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77.b2 77b1 $$[0, 1, 1, -49, 600]$$ $$-13278380032/156590819$$ $$-156590819$$ $$[3]$$ $$20$$ $$0.25431$$ $$\Gamma_0(N)$$-optimal
77.b3 77b2 $$[0, 1, 1, 441, -15815]$$ $$9463555063808/115539436859$$ $$-115539436859$$ $$[]$$ $$60$$ $$0.80361$$
77.b1 77b3 $$[0, 1, 1, -89, 295]$$ $$-78843215872/539$$ $$-539$$ $$[3]$$ $$60$$ $$-0.29500$$

## Rank

sage: E.rank()

The elliptic curves in class 77b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 77b do not have complex multiplication.

## Modular form77.2.a.b

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{4} + 3 q^{5} + q^{7} - 2 q^{9} - q^{11} - 2 q^{12} - 4 q^{13} + 3 q^{15} + 4 q^{16} - 6 q^{17} + 2 q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.