# Properties

 Label 32856.g Number of curves $6$ Conductor $32856$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 32856.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32856.g1 32856c6 $$[0, -1, 0, -526152, -146722068]$$ $$3065617154/9$$ $$47291469170688$$ $$[2]$$ $$202752$$ $$1.8533$$
32856.g2 32856c4 $$[0, -1, 0, -88072, 10088668]$$ $$28756228/3$$ $$7881911528448$$ $$[2]$$ $$101376$$ $$1.5067$$
32856.g3 32856c3 $$[0, -1, 0, -33312, -2221380]$$ $$1556068/81$$ $$212811611268096$$ $$[2, 2]$$ $$101376$$ $$1.5067$$
32856.g4 32856c2 $$[0, -1, 0, -5932, 133300]$$ $$35152/9$$ $$5911433646336$$ $$[2, 2]$$ $$50688$$ $$1.1601$$
32856.g5 32856c1 $$[0, -1, 0, 913, 12828]$$ $$2048/3$$ $$-123154867632$$ $$[2]$$ $$25344$$ $$0.81353$$ $$\Gamma_0(N)$$-optimal
32856.g6 32856c5 $$[0, -1, 0, 21448, -8858292]$$ $$207646/6561$$ $$-34475481025431552$$ $$[2]$$ $$202752$$ $$1.8533$$

## Rank

sage: E.rank()

The elliptic curves in class 32856.g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 32856.g do not have complex multiplication.

## Modular form 32856.2.a.g

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{13} - 2q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.