# Properties

 Label 147.c Number of curves $2$ Conductor $147$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath

This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.

E = EllipticCurve("c1")

E.isogeny_class()

## Elliptic curves in class 147.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
147.c1 147b2 $$[0, 1, 1, -44704, -3655907]$$ $$-1713910976512/1594323$$ $$-9190954824723$$ $$[]$$ $$546$$ $$1.4103$$
147.c2 147b1 $$[0, 1, 1, -114, 473]$$ $$-28672/3$$ $$-17294403$$ $$[]$$ $$42$$ $$0.12787$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 147.c have rank $$0$$.

## Complex multiplication

The elliptic curves in class 147.c do not have complex multiplication.

## Modular form147.2.a.c

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.