# Properties

 Label 397800dx Number of curves $4$ Conductor $397800$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 397800dx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.dx3 397800dx1 $$[0, 0, 0, -7050450, -7205635375]$$ $$212670222886967296/616241925$$ $$112310090831250000$$ $$$$ $$9437184$$ $$2.5017$$ $$\Gamma_0(N)$$-optimal
397800.dx2 397800dx2 $$[0, 0, 0, -7141575, -7009807750]$$ $$13813960087661776/714574355625$$ $$2083698821002500000000$$ $$[2, 2]$$ $$18874368$$ $$2.8483$$
397800.dx1 397800dx3 $$[0, 0, 0, -20304075, 26251829750]$$ $$79364416584061444/20404090514925$$ $$237993311766085200000000$$ $$$$ $$37748736$$ $$3.1949$$
397800.dx4 397800dx4 $$[0, 0, 0, 4562925, -27738477250]$$ $$900753985478876/29018422265625$$ $$-338470877306250000000000$$ $$$$ $$37748736$$ $$3.1949$$

## Rank

sage: E.rank()

The elliptic curves in class 397800dx have rank $$1$$.

## Complex multiplication

The elliptic curves in class 397800dx do not have complex multiplication.

## Modular form 397800.2.a.dx

sage: E.q_eigenform(10)

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