# Properties

 Label 17a Number of curves $4$ Conductor $17$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 17a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17.a3 17a1 $$[1, -1, 1, -1, -14]$$ $$-35937/83521$$ $$-83521$$ $$$$ $$1$$ $$-0.37664$$ $$\Gamma_0(N)$$-optimal
17.a2 17a2 $$[1, -1, 1, -6, -4]$$ $$20346417/289$$ $$289$$ $$[2, 2]$$ $$2$$ $$-0.72321$$
17.a1 17a3 $$[1, -1, 1, -91, -310]$$ $$82483294977/17$$ $$17$$ $$$$ $$4$$ $$-0.37664$$
17.a4 17a4 $$[1, -1, 1, -1, 0]$$ $$35937/17$$ $$17$$ $$$$ $$4$$ $$-1.0698$$

## Rank

sage: E.rank()

The elliptic curves in class 17a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 17a do not have complex multiplication.

## Modular form17.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + q^{17} + 3q^{18} - 4q^{19} + O(q^{20})$$ ## 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. 