# Properties

 Label 50400dx Number of curves $4$ Conductor $50400$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dx1")

sage: E.isogeny_class()

## Elliptic curves in class 50400dx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
50400.cg3 50400dx1 $$[0, 0, 0, -15825, -754000]$$ $$601211584/11025$$ $$8037225000000$$ $$[2, 2]$$ $$147456$$ $$1.2710$$ $$\Gamma_0(N)$$-optimal
50400.cg4 50400dx2 $$[0, 0, 0, -75, -2187250]$$ $$-8/354375$$ $$-2066715000000000$$ $$$$ $$294912$$ $$1.6176$$
50400.cg2 50400dx3 $$[0, 0, 0, -32700, 1136000]$$ $$82881856/36015$$ $$1680315840000000$$ $$$$ $$294912$$ $$1.6176$$
50400.cg1 50400dx4 $$[0, 0, 0, -252075, -48712750]$$ $$303735479048/105$$ $$612360000000$$ $$$$ $$294912$$ $$1.6176$$

## Rank

sage: E.rank()

The elliptic curves in class 50400dx have rank $$2$$.

## Complex multiplication

The elliptic curves in class 50400dx do not have complex multiplication.

## Modular form 50400.2.a.dx

sage: E.q_eigenform(10)

$$q + q^{7} - 4q^{11} - 6q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 