# Properties

 Label 12870p Number of curves 8 Conductor 12870 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("12870.c1")
sage: E.isogeny_class()

## Elliptic curves in class 12870p

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
12870.c7 12870p1 [1, -1, 0, -102780, 12157776] 2 110592 $$\Gamma_0(N)$$-optimal
12870.c6 12870p2 [1, -1, 0, -287100, -43470000] 4 221184
12870.c3 12870p3 [1, -1, 0, -8224380, 9080321616] 6 331776
12870.c5 12870p4 [1, -1, 0, -4247100, -3367494000] 2 442368
12870.c8 12870p5 [1, -1, 0, 723780, -281835504] 2 442368
12870.c2 12870p6 [1, -1, 0, -8225100, 9078652800] 12 663552
12870.c1 12870p7 [1, -1, 0, -9423000, 6261910740] 6 1327104
12870.c4 12870p8 [1, -1, 0, -7038720, 11788581996] 6 1327104

## Rank

sage: E.rank()

The elliptic curves in class 12870p have rank $$0$$.

## Modular form 12870.2.a.c

sage: E.q_eigenform(10)
$$q - q^{2} + q^{4} - q^{5} - 4q^{7} - q^{8} + q^{10} + q^{11} + q^{13} + 4q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.