# Properties

 Label 7728.o Number of curves $6$ Conductor $7728$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 7728.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.o1 7728r5 $$[0, 1, 0, -1266144, 547941492]$$ $$54804145548726848737/637608031452$$ $$2611642496827392$$ $$[4]$$ $$98304$$ $$2.1091$$
7728.o2 7728r3 $$[0, 1, 0, -283424, -58169868]$$ $$614716917569296417/19093020912$$ $$78205013655552$$ $$[2]$$ $$49152$$ $$1.7625$$
7728.o3 7728r4 $$[0, 1, 0, -81184, 8073716]$$ $$14447092394873377/1439452851984$$ $$5895998881726464$$ $$[2, 4]$$ $$49152$$ $$1.7625$$
7728.o4 7728r2 $$[0, 1, 0, -18464, -832524]$$ $$169967019783457/26337394944$$ $$107877969690624$$ $$[2, 2]$$ $$24576$$ $$1.4160$$
7728.o5 7728r1 $$[0, 1, 0, 2016, -70668]$$ $$221115865823/664731648$$ $$-2722740830208$$ $$[2]$$ $$12288$$ $$1.0694$$ $$\Gamma_0(N)$$-optimal
7728.o6 7728r6 $$[0, 1, 0, 100256, 39208820]$$ $$27207619911317663/177609314617308$$ $$-727487752672493568$$ $$[4]$$ $$98304$$ $$2.1091$$

## Rank

sage: E.rank()

The elliptic curves in class 7728.o have rank $$0$$.

## Complex multiplication

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

## Modular form7728.2.a.o

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} - q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.