# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5415k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5415.j7 5415k1 $$[1, 0, 1, -8, -1279]$$ $$-1/15$$ $$-705688215$$ $$[2]$$ $$1728$$ $$0.37679$$ $$\Gamma_0(N)$$-optimal
5415.j6 5415k2 $$[1, 0, 1, -1813, -29437]$$ $$13997521/225$$ $$10585323225$$ $$[2, 2]$$ $$3456$$ $$0.72337$$
5415.j4 5415k3 $$[1, 0, 1, -28888, -1892197]$$ $$56667352321/15$$ $$705688215$$ $$[2]$$ $$6912$$ $$1.0699$$
5415.j5 5415k4 $$[1, 0, 1, -3618, 38431]$$ $$111284641/50625$$ $$2381697725625$$ $$[2, 2]$$ $$6912$$ $$1.0699$$
5415.j2 5415k5 $$[1, 0, 1, -48743, 4135781]$$ $$272223782641/164025$$ $$7716700631025$$ $$[2, 2]$$ $$13824$$ $$1.4165$$
5415.j8 5415k6 $$[1, 0, 1, 12627, 291853]$$ $$4733169839/3515625$$ $$-165395675390625$$ $$[2]$$ $$13824$$ $$1.4165$$
5415.j1 5415k7 $$[1, 0, 1, -779768, 264965501]$$ $$1114544804970241/405$$ $$19053581805$$ $$[2]$$ $$27648$$ $$1.7631$$
5415.j3 5415k8 $$[1, 0, 1, -39718, 5716961]$$ $$-147281603041/215233605$$ $$-10125854568031005$$ $$[2]$$ $$27648$$ $$1.7631$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 5415k do not have complex multiplication.

## Modular form5415.2.a.k

sage: E.q_eigenform(10)

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

## 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.