# Properties

 Label 117600de Number of curves $4$ Conductor $117600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("de1")

sage: E.isogeny_class()

## Elliptic curves in class 117600de

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117600.hq3 117600de1 $$[0, 1, 0, -7813969158, 265858972088688]$$ $$448487713888272974160064/91549016015625$$ $$10770650185222265625000000$$ $$[2, 2]$$ $$123863040$$ $$4.1911$$ $$\Gamma_0(N)$$-optimal
117600.hq4 117600de2 $$[0, 1, 0, -7787178408, 267772528198188]$$ $$-55486311952875723077768/801237030029296875$$ $$-754117882767333984375000000000$$ $$[2]$$ $$247726080$$ $$4.5377$$
117600.hq2 117600de3 $$[0, 1, 0, -7840766033, 263943665448063]$$ $$7079962908642659949376/100085966990454375$$ $$753600891549437872920000000000$$ $$[2]$$ $$247726080$$ $$4.5377$$
117600.hq1 117600de4 $$[0, 1, 0, -125023500408, 17015100987713688]$$ $$229625675762164624948320008/9568125$$ $$9005442705000000000$$ $$[2]$$ $$247726080$$ $$4.5377$$

## Rank

sage: E.rank()

The elliptic curves in class 117600de have rank $$1$$.

## Complex multiplication

The elliptic curves in class 117600de do not have complex multiplication.

## Modular form 117600.2.a.de

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} + 4q^{11} - 6q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.