# Properties

 Label 50400df Number of curves $4$ Conductor $50400$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

## Elliptic curves in class 50400df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
50400.ca3 50400df1 $$[0, 0, 0, -1435218825, -20927879219000]$$ $$448487713888272974160064/91549016015625$$ $$66739232675390625000000$$ $$[2, 2]$$ $$20643840$$ $$3.7674$$ $$\Gamma_0(N)$$-optimal
50400.ca4 50400df2 $$[0, 0, 0, -1430298075, -21078508297250]$$ $$-55486311952875723077768/801237030029296875$$ $$-4672814359130859375000000000$$ $$[2]$$ $$41287680$$ $$4.1140$$
50400.ca2 50400df3 $$[0, 0, 0, -1440140700, -20777112344000]$$ $$7079962908642659949376/100085966990454375$$ $$4669610875906639320000000000$$ $$[2]$$ $$41287680$$ $$4.1140$$
50400.ca1 50400df4 $$[0, 0, 0, -22963500075, -1339384407812750]$$ $$229625675762164624948320008/9568125$$ $$55801305000000000$$ $$[2]$$ $$41287680$$ $$4.1140$$

## Rank

sage: E.rank()

The elliptic curves in class 50400df have rank $$1$$.

## Complex multiplication

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

## Modular form 50400.2.a.df

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.