# Properties

 Label 5415k Number of curves 8 Conductor 5415 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5415.j1")

sage: E.isogeny_class()

## Elliptic curves in class 5415k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5415.j7 5415k1 [1, 0, 1, -8, -1279] [2] 1728 $$\Gamma_0(N)$$-optimal
5415.j6 5415k2 [1, 0, 1, -1813, -29437] [2, 2] 3456
5415.j4 5415k3 [1, 0, 1, -28888, -1892197] [2] 6912
5415.j5 5415k4 [1, 0, 1, -3618, 38431] [2, 2] 6912
5415.j2 5415k5 [1, 0, 1, -48743, 4135781] [2, 2] 13824
5415.j8 5415k6 [1, 0, 1, 12627, 291853] [2] 13824
5415.j1 5415k7 [1, 0, 1, -779768, 264965501] [2] 27648
5415.j3 5415k8 [1, 0, 1, -39718, 5716961] [2] 27648

## Rank

sage: E.rank()

The elliptic curves in class 5415k have rank $$0$$.

## Modular form5415.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3q^{8} + q^{9} + q^{10} - 4q^{11} - q^{12} + 2q^{13} + q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.