# Properties

 Label 212160z Number of curves 8 Conductor 212160 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("212160.hq1")

sage: E.isogeny_class()

## Elliptic curves in class 212160z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
212160.hq7 212160z1 [0, 1, 0, -134153345, -575935024257] [2] 53084160 $$\Gamma_0(N)$$-optimal
212160.hq6 212160z2 [0, 1, 0, -355665025, 1815815189375] [2, 2] 106168320
212160.hq5 212160z3 [0, 1, 0, -1650738305, 25645921710975] [2] 159252480
212160.hq4 212160z4 [0, 1, 0, -5207889025, 144637146850175] [2] 212336640
212160.hq8 212160z5 [0, 1, 0, 952372095, 12070041387903] [2] 212336640
212160.hq2 212160z6 [0, 1, 0, -26364000385, 1647641566502783] [2, 2] 318504960
212160.hq1 212160z7 [0, 1, 0, -421824000385, 105449485114502783] [2] 637009920
212160.hq3 212160z8 [0, 1, 0, -26316193665, 1653914697371775] [2] 637009920

## Rank

sage: E.rank()

The elliptic curves in class 212160z have rank $$1$$.

## Modular form 212160.2.a.hq

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.