# Properties

 Label 315.a Number of curves $4$ Conductor $315$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 315.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
315.a1 315b3 [1, -1, 1, -1013, 12656] [2] 128
315.a2 315b2 [1, -1, 1, -68, 182] [2, 2] 64
315.a3 315b1 [1, -1, 1, -23, -34] [2] 32 $$\Gamma_0(N)$$-optimal
315.a4 315b4 [1, -1, 1, 157, 992] [2] 128

## Rank

sage: E.rank()

The elliptic curves in class 315.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 315.a do not have complex multiplication.

## Modular form315.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - q^{5} + q^{7} + 3q^{8} + q^{10} - 6q^{13} - q^{14} - q^{16} - 2q^{17} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.