# Properties

 Label 212160hl Number of curves $6$ Conductor $212160$ CM no Rank $2$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("hl1")

sage: E.isogeny_class()

## Elliptic curves in class 212160hl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.x4 212160hl1 $$[0, -1, 0, -66301, 6593101]$$ $$31476797652269056/49725$$ $$50918400$$ $$[2]$$ $$360448$$ $$1.1732$$ $$\Gamma_0(N)$$-optimal
212160.x3 212160hl2 $$[0, -1, 0, -66321, 6588945]$$ $$1969080716416336/2472575625$$ $$40510679040000$$ $$[2, 2]$$ $$720896$$ $$1.5198$$
212160.x5 212160hl3 $$[0, -1, 0, -48641, 10163841]$$ $$-194204905090564/566398828125$$ $$-37119513600000000$$ $$[2]$$ $$1441792$$ $$1.8664$$
212160.x2 212160hl4 $$[0, -1, 0, -84321, 2747745]$$ $$1011710313226084/536724738225$$ $$35174792444313600$$ $$[2, 2]$$ $$1441792$$ $$1.8664$$
212160.x6 212160hl5 $$[0, -1, 0, 321279, 21161985]$$ $$27980756504588158/17683545112935$$ $$-2317817625042616320$$ $$[2]$$ $$2883584$$ $$2.2129$$
212160.x1 212160hl6 $$[0, -1, 0, -777921, -261791295]$$ $$397210600760070242/3536192675535$$ $$463495846367723520$$ $$[2]$$ $$2883584$$ $$2.2129$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 212160.2.a.hl

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} - 4q^{11} - q^{13} + q^{15} + q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.