# Properties

 Label 172a Number of curves 2 Conductor 172 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("172.a1")
sage: E.isogeny_class()

## Elliptic curves in class 172a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
172.a1 172a1 [0, 1, 0, -13, 15] 3 12 $$\Gamma_0(N)$$-optimal
172.a2 172a2 [0, 1, 0, 67, 79] 1 36

## Rank

sage: E.rank()

The elliptic curves in class 172a have rank $$1$$.

## Modular form172.2.a.a

sage: E.q_eigenform(10)
$$q - 2q^{3} - 4q^{7} + q^{9} - 3q^{11} - q^{13} - 3q^{17} + 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 