# Properties

 Label 770.f Number of curves $4$ Conductor $770$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 770.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.f1 770f4 $$[1, 0, 0, -26116, -1580604]$$ $$1969902499564819009/63690429687500$$ $$63690429687500$$ $$$$ $$3456$$ $$1.4227$$
770.f2 770f2 $$[1, 0, 0, -3576, 81280]$$ $$5057359576472449/51765560000$$ $$51765560000$$ $$$$ $$1152$$ $$0.87336$$
770.f3 770f1 $$[1, 0, 0, -56, 3136]$$ $$-19443408769/4249907200$$ $$-4249907200$$ $$$$ $$576$$ $$0.52678$$ $$\Gamma_0(N)$$-optimal
770.f4 770f3 $$[1, 0, 0, 504, -84560]$$ $$14156681599871/3100231750000$$ $$-3100231750000$$ $$$$ $$1728$$ $$1.0761$$

## Rank

sage: E.rank()

The elliptic curves in class 770.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 770.f do not have complex multiplication.

## Modular form770.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{11} - 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 