# Properties

 Label 356928fa Number of curves $6$ Conductor $356928$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("356928.fa1")

sage: E.isogeny_class()

## Elliptic curves in class 356928fa

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
356928.fa5 356928fa1 [0, 1, 0, -259809, -73204065] [2] 5505024 $$\Gamma_0(N)$$-optimal
356928.fa4 356928fa2 [0, 1, 0, -4640289, -3848301729] [2, 2] 11010048
356928.fa3 356928fa3 [0, 1, 0, -5127009, -2992161249] [2, 2] 22020096
356928.fa1 356928fa4 [0, 1, 0, -74241249, -246240605025] [2] 22020096
356928.fa2 356928fa5 [0, 1, 0, -32545569, 69189939807] [2] 44040192
356928.fa6 356928fa6 [0, 1, 0, 14504031, -20373484065] [2] 44040192

## Rank

sage: E.rank()

The elliptic curves in class 356928fa have rank $$0$$.

## Modular form 356928.2.a.fa

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} + q^{11} - 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.