# Properties

 Label 112710cr Number of curves $6$ Conductor $112710$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("112710.cn1")

sage: E.isogeny_class()

## Elliptic curves in class 112710cr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112710.cn6 112710cr1 [1, 0, 0, 4329, 44265] [2] 294912 $$\Gamma_0(N)$$-optimal
112710.cn5 112710cr2 [1, 0, 0, -18791, 363321] [2, 2] 589824
112710.cn3 112710cr3 [1, 0, 0, -163291, -25155379] [2, 2] 1179648
112710.cn2 112710cr4 [1, 0, 0, -244211, 46394085] [2] 1179648
112710.cn4 112710cr5 [1, 0, 0, -33241, -64092349] [2] 2359296
112710.cn1 112710cr6 [1, 0, 0, -2605341, -1618837209] [2] 2359296

## Rank

sage: E.rank()

The elliptic curves in class 112710cr have rank $$0$$.

## Modular form 112710.2.a.cn

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + q^{13} - q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.