# Properties

 Label 15210n Number of curves 8 Conductor 15210 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("15210.k1")

sage: E.isogeny_class()

## Elliptic curves in class 15210n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15210.k8 15210n1 [1, -1, 0, 2250, -126684] [2] 36864 $$\Gamma_0(N)$$-optimal
15210.k6 15210n2 [1, -1, 0, -28170, -1641600] [2, 2] 73728
15210.k7 15210n3 [1, -1, 0, -20565, 3719925] [2] 110592
15210.k4 15210n4 [1, -1, 0, -438840, -111783294] [2] 147456
15210.k5 15210n5 [1, -1, 0, -104220, 11180430] [2] 147456
15210.k3 15210n6 [1, -1, 0, -507285, 138930741] [2, 2] 221184
15210.k2 15210n7 [1, -1, 0, -689805, 30185325] [2] 442368
15210.k1 15210n8 [1, -1, 0, -8112285, 8895327741] [2] 442368

## Rank

sage: E.rank()

The elliptic curves in class 15210n have rank $$0$$.

## Modular form 15210.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} + 4q^{7} - q^{8} + q^{10} - 4q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.