# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("479370.z1")

sage: E.isogeny_class()

## Elliptic curves in class 479370z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
479370.z3 479370z1 [1, 0, 1, -26089, 1610012]  1548288 $$\Gamma_0(N)$$-optimal
479370.z2 479370z2 [1, 0, 1, -42909, -724604] [2, 2] 3096576
479370.z4 479370z3 [1, 0, 1, 167341, -5686504]  6193152
479370.z1 479370z4 [1, 0, 1, -522279, -145110848]  6193152

## Rank

sage: E.rank()

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

## 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 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. 