# Properties

 Label 6720cg Number of curves $8$ Conductor $6720$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6720.cc1")

sage: E.isogeny_class()

## Elliptic curves in class 6720cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.cc7 6720cg1 [0, 1, 0, -2625, 17343] [2] 9216 $$\Gamma_0(N)$$-optimal
6720.cc5 6720cg2 [0, 1, 0, -23105, -1346625] [2, 2] 18432
6720.cc4 6720cg3 [0, 1, 0, -171585, 27299775] [2] 27648
6720.cc2 6720cg4 [0, 1, 0, -368705, -86295105] [2] 36864
6720.cc6 6720cg5 [0, 1, 0, -5185, -3364417] [4] 36864
6720.cc3 6720cg6 [0, 1, 0, -172865, 26870463] [2, 2] 55296
6720.cc1 6720cg7 [0, 1, 0, -412865, -64377537] [2] 110592
6720.cc8 6720cg8 [0, 1, 0, 46655, 90662975] [4] 110592

## Rank

sage: E.rank()

The elliptic curves in class 6720cg have rank $$0$$.

## Modular form6720.2.a.cc

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.