# Properties

 Label 479370.z Number of curves $4$ Conductor $479370$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 479370.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
479370.z1 479370z4 $$[1, 0, 1, -522279, -145110848]$$ $$26487576322129/44531250$$ $$26488226013281250$$ $$$$ $$6193152$$ $$2.0466$$
479370.z2 479370z2 $$[1, 0, 1, -42909, -724604]$$ $$14688124849/8122500$$ $$4831452424822500$$ $$[2, 2]$$ $$3096576$$ $$1.7000$$
479370.z3 479370z1 $$[1, 0, 1, -26089, 1610012]$$ $$3301293169/22800$$ $$13561971718800$$ $$$$ $$1548288$$ $$1.3534$$ $$\Gamma_0(N)$$-optimal*
479370.z4 479370z3 $$[1, 0, 1, 167341, -5686504]$$ $$871257511151/527800050$$ $$-313947778564966050$$ $$$$ $$6193152$$ $$2.0466$$
*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 479370.z1.

## Rank

sage: E.rank()

The elliptic curves in class 479370.z have rank $$1$$.

## Complex multiplication

The elliptic curves in class 479370.z do not have complex multiplication.

## Modular form 479370.2.a.z

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} + 2q^{13} - q^{15} + q^{16} - 2q^{17} - q^{18} + q^{19} + O(q^{20})$$ ## 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. 