# Properties

 Label 15.a Number of curves $8$ Conductor $15$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 15.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15.a1 15a5 $$[1, 1, 1, -2160, -39540]$$ $$1114544804970241/405$$ $$405$$ $$[2]$$ $$4$$ $$0.29087$$
15.a2 15a2 $$[1, 1, 1, -135, -660]$$ $$272223782641/164025$$ $$164025$$ $$[2, 2]$$ $$2$$ $$-0.055704$$
15.a3 15a6 $$[1, 1, 1, -110, -880]$$ $$-147281603041/215233605$$ $$-215233605$$ $$[2]$$ $$4$$ $$0.29087$$
15.a4 15a7 $$[1, 1, 1, -80, 242]$$ $$56667352321/15$$ $$15$$ $$[4]$$ $$4$$ $$-0.40228$$
15.a5 15a1 $$[1, 1, 1, -10, -10]$$ $$111284641/50625$$ $$50625$$ $$[2, 4]$$ $$1$$ $$-0.40228$$ $$\Gamma_0(N)$$-optimal
15.a6 15a3 $$[1, 1, 1, -5, 2]$$ $$13997521/225$$ $$225$$ $$[2, 4]$$ $$2$$ $$-0.74885$$
15.a7 15a8 $$[1, 1, 1, 0, 0]$$ $$-1/15$$ $$-15$$ $$[4]$$ $$4$$ $$-1.0954$$
15.a8 15a4 $$[1, 1, 1, 35, -28]$$ $$4733169839/3515625$$ $$-3515625$$ $$[8]$$ $$2$$ $$-0.055704$$

## Rank

sage: E.rank()

The elliptic curves in class 15.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 15.a do not have complex multiplication.

## Modular form15.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.