# Properties

 Label 212160hd Number of curves $4$ Conductor $212160$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("hd1")

sage: E.isogeny_class()

## Elliptic curves in class 212160hd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.d3 212160hd1 $$[0, -1, 0, -125341, 17121805]$$ $$212670222886967296/616241925$$ $$631031731200$$ $$$$ $$786432$$ $$1.4943$$ $$\Gamma_0(N)$$-optimal
212160.d2 212160hd2 $$[0, -1, 0, -126961, 16658161]$$ $$13813960087661776/714574355625$$ $$11707586242560000$$ $$[2, 2]$$ $$1572864$$ $$1.8409$$
212160.d4 212160hd3 $$[0, -1, 0, 81119, 65723425]$$ $$900753985478876/29018422265625$$ $$-1901751321600000000$$ $$$$ $$3145728$$ $$2.1874$$
212160.d1 212160hd4 $$[0, -1, 0, -360961, -62106239]$$ $$79364416584061444/20404090514925$$ $$1337202475986124800$$ $$$$ $$3145728$$ $$2.1874$$

## Rank

sage: E.rank()

The elliptic curves in class 212160hd have rank $$2$$.

## Complex multiplication

The elliptic curves in class 212160hd do not have complex multiplication.

## Modular form 212160.2.a.hd

sage: E.q_eigenform(10)

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