# Properties

 Label 315a Number of curves $3$ Conductor $315$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 315a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
315.b2 315a1 $$[0, 0, 1, -12, -18]$$ $$-262144/35$$ $$-25515$$ $$[]$$ $$20$$ $$-0.42184$$ $$\Gamma_0(N)$$-optimal
315.b3 315a2 $$[0, 0, 1, 78, 45]$$ $$71991296/42875$$ $$-31255875$$ $$$$ $$60$$ $$0.12746$$
315.b1 315a3 $$[0, 0, 1, -1182, 16362]$$ $$-250523582464/13671875$$ $$-9966796875$$ $$$$ $$180$$ $$0.67677$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form315.2.a.a

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 