# Properties

 Label 494190.v Number of curves $4$ Conductor $494190$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("v1")

sage: E.isogeny_class()

## Elliptic curves in class 494190.v

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
494190.v1 494190v3 $$[1, -1, 0, -1615275, 789420411]$$ $$26487576322129/44531250$$ $$783584691138281250$$ $$[2]$$ $$10485760$$ $$2.3288$$ $$\Gamma_0(N)$$-optimal*
494190.v2 494190v2 $$[1, -1, 0, -132705, 3954825]$$ $$14688124849/8122500$$ $$142925847663622500$$ $$[2, 2]$$ $$5242880$$ $$1.9823$$ $$\Gamma_0(N)$$-optimal*
494190.v3 494190v1 $$[1, -1, 0, -80685, -8748459]$$ $$3301293169/22800$$ $$401195361862800$$ $$[2]$$ $$2621440$$ $$1.6357$$ $$\Gamma_0(N)$$-optimal*
494190.v4 494190v4 $$[1, -1, 0, 517545, 30875175]$$ $$871257511151/527800050$$ $$-9287321581182190050$$ $$[2]$$ $$10485760$$ $$2.3288$$
*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 494190.v1.

## Rank

sage: E.rank()

The elliptic curves in class 494190.v have rank $$0$$.

## Complex multiplication

The elliptic curves in class 494190.v do not have complex multiplication.

## Modular form 494190.2.a.v

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + 4q^{11} + 2q^{13} + q^{16} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.