# Properties

 Label 7728.a Number of curves $2$ Conductor $7728$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 7728.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.a1 7728j2 $$[0, -1, 0, -432, 15936]$$ $$-2181825073/25039686$$ $$-102562553856$$ $$[]$$ $$8640$$ $$0.79475$$
7728.a2 7728j1 $$[0, -1, 0, 48, -576]$$ $$2924207/34776$$ $$-142442496$$ $$[]$$ $$2880$$ $$0.24544$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 7728.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 7728.a do not have complex multiplication.

## Modular form7728.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - 3 q^{5} - q^{7} + q^{9} + 5 q^{13} + 3 q^{15} - 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.