# Properties

 Label 315.b Number of curves 3 Conductor 315 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("315.b1")

sage: E.isogeny_class()

## Elliptic curves in class 315.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
315.b1 315a3 [0, 0, 1, -1182, 16362]  180
315.b2 315a1 [0, 0, 1, -12, -18] [] 20 $$\Gamma_0(N)$$-optimal
315.b3 315a2 [0, 0, 1, 78, 45]  60

## Rank

sage: E.rank()

The elliptic curves in class 315.b have rank $$0$$.

## Modular form315.2.a.b

sage: E.q_eigenform(10)

$$q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 