# Properties

 Label 5070w Number of curves $4$ Conductor $5070$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 5070w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5070.u4 5070w1 $$[1, 0, 0, -48760, 6842432]$$ $$-2656166199049/2658140160$$ $$-12830334847549440$$ $$$$ $$53760$$ $$1.7866$$ $$\Gamma_0(N)$$-optimal
5070.u3 5070w2 $$[1, 0, 0, -914040, 336168000]$$ $$17496824387403529/6580454400$$ $$31762596522009600$$ $$[2, 2]$$ $$107520$$ $$2.1332$$
5070.u2 5070w3 $$[1, 0, 0, -1049240, 230144160]$$ $$26465989780414729/10571870144160$$ $$51028397958662785440$$ $$$$ $$215040$$ $$2.4797$$
5070.u1 5070w4 $$[1, 0, 0, -14623320, 21522489312]$$ $$71647584155243142409/10140000$$ $$48943843260000$$ $$$$ $$215040$$ $$2.4797$$

## Rank

sage: E.rank()

The elliptic curves in class 5070w have rank $$0$$.

## Complex multiplication

The elliptic curves in class 5070w do not have complex multiplication.

## Modular form5070.2.a.w

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 4q^{14} + q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$ ## 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. 