# Properties

 Label 7098.k Number of curves $4$ Conductor $7098$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 7098.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7098.k1 7098o3 $$[1, 0, 1, -2284377, 1328255164]$$ $$124318741396429/51631104$$ $$547522009995297792$$ $$[2]$$ $$156000$$ $$2.3655$$
7098.k2 7098o4 $$[1, 0, 1, -1932857, 1751063420]$$ $$-75306487574989/81352871712$$ $$-862706477061653436576$$ $$[2]$$ $$312000$$ $$2.7121$$
7098.k3 7098o1 $$[1, 0, 1, -76392, -8122814]$$ $$4649101309/6804$$ $$72153013733892$$ $$[2]$$ $$31200$$ $$1.5608$$ $$\Gamma_0(N)$$-optimal
7098.k4 7098o2 $$[1, 0, 1, -54422, -12885910]$$ $$-1680914269/5786802$$ $$-61366138180675146$$ $$[2]$$ $$62400$$ $$1.9074$$

## Rank

sage: E.rank()

The elliptic curves in class 7098.k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 7098.k do not have complex multiplication.

## Modular form7098.2.a.k

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.