# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("7728.o1")

sage: E.isogeny_class()

## Elliptic curves in class 7728r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
7728.o5 7728r1 [0, 1, 0, 2016, -70668] [2] 12288 $$\Gamma_0(N)$$-optimal
7728.o4 7728r2 [0, 1, 0, -18464, -832524] [2, 2] 24576
7728.o2 7728r3 [0, 1, 0, -283424, -58169868] [2] 49152
7728.o3 7728r4 [0, 1, 0, -81184, 8073716] [2, 4] 49152
7728.o1 7728r5 [0, 1, 0, -1266144, 547941492] [4] 98304
7728.o6 7728r6 [0, 1, 0, 100256, 39208820] [4] 98304

## Rank

sage: E.rank()

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

## Modular form7728.2.a.o

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.