# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("gl1")

sage: E.isogeny_class()

## Elliptic curves in class 212160gl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.cx3 212160gl1 $$[0, -1, 0, -1905, 28017]$$ $$46689225424/7249905$$ $$118782443520$$ $$$$ $$196608$$ $$0.84829$$ $$\Gamma_0(N)$$-optimal
212160.cx2 212160gl2 $$[0, -1, 0, -8385, -266175]$$ $$994958062276/98903025$$ $$6481708646400$$ $$[2, 2]$$ $$393216$$ $$1.1949$$
212160.cx4 212160gl3 $$[0, -1, 0, 10335, -1303263]$$ $$931329171502/6107473125$$ $$-800518717440000$$ $$$$ $$786432$$ $$1.5414$$
212160.cx1 212160gl4 $$[0, -1, 0, -130785, -18161055]$$ $$1887517194957938/21849165$$ $$2863813754880$$ $$$$ $$786432$$ $$1.5414$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 212160.2.a.gl

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 4 q^{11} + q^{13} - q^{15} + q^{17} - 8 q^{19} + O(q^{20})$$

## 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. 