# Properties

 Label 405720.gm Number of curves $2$ Conductor $405720$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("gm1")

sage: E.isogeny_class()

## Elliptic curves in class 405720.gm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405720.gm1 405720gm1 $$[0, 0, 0, -38367, -796446]$$ $$10536048/5635$$ $$3340514938763520$$ $$$$ $$2211840$$ $$1.6700$$ $$\Gamma_0(N)$$-optimal
405720.gm2 405720gm2 $$[0, 0, 0, 146853, -6241914]$$ $$147704148/92575$$ $$-219519553118745600$$ $$$$ $$4423680$$ $$2.0166$$

## Rank

sage: E.rank()

The elliptic curves in class 405720.gm have rank $$0$$.

## Complex multiplication

The elliptic curves in class 405720.gm do not have complex multiplication.

## Modular form 405720.2.a.gm

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{11} - 4q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 