# Properties

 Label 847.a Number of curves $2$ Conductor $847$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 847.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
847.a1 847c2 $$[1, 1, 1, -6234, -177484]$$ $$15124197817/1294139$$ $$2292646180979$$ $$$$ $$1440$$ $$1.1129$$
847.a2 847c1 $$[1, 1, 1, 421, -12440]$$ $$4657463/41503$$ $$-73525096183$$ $$$$ $$720$$ $$0.76635$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 847.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 847.a do not have complex multiplication.

## Modular form847.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} + q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} - q^{14} - 4 q^{15} - q^{16} - 4 q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 