# Properties

 Label 481338df Number of curves $4$ Conductor $481338$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 481338df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481338.df4 481338df1 $$[1, -1, 1, 975020881, -1286710394570697]$$ $$79374649975090937760383/553856914190911653543936$$ $$-715288464086743951460818678185984$$ $$[2]$$ $$2269347840$$ $$4.9831$$ $$\Gamma_0(N)$$-optimal*
481338.df3 481338df2 $$[1, -1, 1, -171737242799, -26863669522941897]$$ $$433744050935826360922067531137/9612122270219882316693504$$ $$12413748026100540598959174406373376$$ $$[2, 2]$$ $$4538695680$$ $$5.3297$$ $$\Gamma_0(N)$$-optimal*
481338.df2 481338df3 $$[1, -1, 1, -373858430639, 48387423133967415]$$ $$4474676144192042711273397261697/1806328356954994499451382272$$ $$2332815214503773770612648277062445568$$ $$[2]$$ $$9077391360$$ $$5.6763$$ $$\Gamma_0(N)$$-optimal*
481338.df1 481338df4 $$[1, -1, 1, -2733012273839, -1739042218942772169]$$ $$1748094148784980747354970849498497/887694600425282263291392$$ $$1146429142703505820824657281422848$$ $$[2]$$ $$9077391360$$ $$5.6763$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 481338df1.

## Rank

sage: E.rank()

The elliptic curves in class 481338df have rank $$0$$.

## Complex multiplication

The elliptic curves in class 481338df do not have complex multiplication.

## Modular form 481338.2.a.df

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2q^{5} + q^{8} + 2q^{10} - q^{13} + q^{16} - q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.