# Properties

 Label 1710h Number of curves $4$ Conductor $1710$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 1710h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.i3 1710h1 $$[1, -1, 0, -279, -1715]$$ $$3301293169/22800$$ $$16621200$$ $$[2]$$ $$512$$ $$0.21908$$ $$\Gamma_0(N)$$-optimal
1710.i2 1710h2 $$[1, -1, 0, -459, 913]$$ $$14688124849/8122500$$ $$5921302500$$ $$[2, 2]$$ $$1024$$ $$0.56565$$
1710.i1 1710h3 $$[1, -1, 0, -5589, 161995]$$ $$26487576322129/44531250$$ $$32463281250$$ $$[2]$$ $$2048$$ $$0.91222$$
1710.i4 1710h4 $$[1, -1, 0, 1791, 5863]$$ $$871257511151/527800050$$ $$-384766236450$$ $$[2]$$ $$2048$$ $$0.91222$$

## Rank

sage: E.rank()

The elliptic curves in class 1710h have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1710h do not have complex multiplication.

## Modular form1710.2.a.h

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{16} - 2q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.