# Properties

 Label 466752.dt Number of curves $4$ Conductor $466752$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("dt1")

sage: E.isogeny_class()

## Elliptic curves in class 466752.dt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.dt1 466752dt3 $$[0, 1, 0, -210894113, 1178543587935]$$ $$3957101249824708884951625/772310238681366528$$ $$202456495208888147116032$$ $$[2]$$ $$87588864$$ $$3.4718$$ $$\Gamma_0(N)$$-optimal*
466752.dt2 466752dt4 $$[0, 1, 0, -188611873, 1437325066847]$$ $$-2830680648734534916567625/1766676274677722124288$$ $$-463123585349116788549353472$$ $$[2]$$ $$175177728$$ $$3.8184$$
466752.dt3 466752dt1 $$[0, 1, 0, -6420833, -4057229793]$$ $$111675519439697265625/37528570137307392$$ $$9837889490074308968448$$ $$[2]$$ $$29196288$$ $$2.9225$$ $$\Gamma_0(N)$$-optimal*
466752.dt4 466752dt2 $$[0, 1, 0, 18733727, -28019463649]$$ $$2773679829880629422375/2899504554614368272$$ $$-760087721964828956295168$$ $$[2]$$ $$58392576$$ $$3.2691$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 466752.dt1.

## Rank

sage: E.rank()

The elliptic curves in class 466752.dt have rank $$1$$.

## Complex multiplication

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

## Modular form 466752.2.a.dt

sage: E.q_eigenform(10)

$$q + q^{3} + 4 q^{7} + q^{9} + q^{11} - q^{13} + q^{17} + 2 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.