# Properties

 Label 466752.eb Number of curves $4$ Conductor $466752$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("eb1")

sage: E.isogeny_class()

## Elliptic curves in class 466752.eb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.eb1 466752eb4 $$[0, 1, 0, -160617801217, -24776466871606177]$$ $$1748094148784980747354970849498497/887694600425282263291392$$ $$232703813333885193628258664448$$ $$[2]$$ $$1815478272$$ $$4.9677$$
466752.eb2 466752eb3 $$[0, 1, 0, -21971478017, 689374449575007]$$ $$4474676144192042711273397261697/1806328356954994499451382272$$ $$473518140805610078064183154311168$$ $$[2]$$ $$1815478272$$ $$4.9677$$ $$\Gamma_0(N)$$-optimal*
466752.eb3 466752eb2 $$[0, 1, 0, -10092914177, -382734832655265]$$ $$433744050935826360922067531137/9612122270219882316693504$$ $$2519760180404520830027301912576$$ $$[2, 2]$$ $$907739136$$ $$4.6211$$ $$\Gamma_0(N)$$-optimal*
466752.eb4 466752eb1 $$[0, 1, 0, 57301503, -18331939527585]$$ $$79374649975090937760383/553856914190911653543936$$ $$-145190266913662344506621558784$$ $$[2]$$ $$453869568$$ $$4.2746$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 466752.eb1.

## Rank

sage: E.rank()

The elliptic curves in class 466752.eb have rank $$0$$.

## Complex multiplication

The elliptic curves in class 466752.eb do not have complex multiplication.

## Modular form 466752.2.a.eb

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} - q^{11} - q^{13} + 2q^{15} - q^{17} - 8q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.