# Properties

 Label 210e Number of curves $8$ Conductor $210$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 210e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210.e7 210e1 $$[1, 0, 0, 210, 900]$$ $$1023887723039/928972800$$ $$-928972800$$ $$[8]$$ $$128$$ $$0.40813$$ $$\Gamma_0(N)$$-optimal
210.e6 210e2 $$[1, 0, 0, -1070, 7812]$$ $$135487869158881/51438240000$$ $$51438240000$$ $$[2, 8]$$ $$256$$ $$0.75471$$
210.e5 210e3 $$[1, 0, 0, -7550, -247500]$$ $$47595748626367201/1215506250000$$ $$1215506250000$$ $$[2, 4]$$ $$512$$ $$1.1013$$
210.e4 210e4 $$[1, 0, 0, -15070, 710612]$$ $$378499465220294881/120530818800$$ $$120530818800$$ $$[8]$$ $$512$$ $$1.1013$$
210.e2 210e5 $$[1, 0, 0, -120050, -16020000]$$ $$191342053882402567201/129708022500$$ $$129708022500$$ $$[2, 2]$$ $$1024$$ $$1.4479$$
210.e8 210e6 $$[1, 0, 0, 1270, -789048]$$ $$226523624554079/269165039062500$$ $$-269165039062500$$ $$[4]$$ $$1024$$ $$1.4479$$
210.e1 210e7 $$[1, 0, 0, -1920800, -1024800150]$$ $$783736670177727068275201/360150$$ $$360150$$ $$[2]$$ $$2048$$ $$1.7944$$
210.e3 210e8 $$[1, 0, 0, -119300, -16229850]$$ $$-187778242790732059201/4984939585440150$$ $$-4984939585440150$$ $$[2]$$ $$2048$$ $$1.7944$$

## Rank

sage: E.rank()

The elliptic curves in class 210e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 210e do not have complex multiplication.

## Modular form210.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.