# Properties

 Label 20097h Number of curves $4$ Conductor $20097$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 20097h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20097.l4 20097h1 $$[1, -1, 0, 5259, -168616]$$ $$22062729659823/29354283343$$ $$-21399272557047$$ $$[2]$$ $$43008$$ $$1.2437$$ $$\Gamma_0(N)$$-optimal
20097.l3 20097h2 $$[1, -1, 0, -32586, -1629433]$$ $$5249244962308257/1448621666569$$ $$1056045194928801$$ $$[2, 2]$$ $$86016$$ $$1.5903$$
20097.l1 20097h3 $$[1, -1, 0, -480201, -127946386]$$ $$16798320881842096017/2132227789307$$ $$1554394058404803$$ $$[2]$$ $$172032$$ $$1.9368$$
20097.l2 20097h4 $$[1, -1, 0, -190491, 30741092]$$ $$1048626554636928177/48569076788309$$ $$35406856978677261$$ $$[2]$$ $$172032$$ $$1.9368$$

## Rank

sage: E.rank()

The elliptic curves in class 20097h have rank $$1$$.

## Complex multiplication

The elliptic curves in class 20097h do not have complex multiplication.

## Modular form 20097.2.a.h

sage: E.q_eigenform(10)

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

## 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.