# Properties

 Label 147.b Number of curves $2$ Conductor $147$ CM no Rank $0$ Graph # Related objects

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

E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 147.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
147.b1 147c2 $$[0, -1, 1, -912, 10919]$$ $$-1713910976512/1594323$$ $$-78121827$$ $$[]$$ $$78$$ $$0.43739$$
147.b2 147c1 $$[0, -1, 1, -2, -1]$$ $$-28672/3$$ $$-147$$ $$[]$$ $$6$$ $$-0.84509$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form147.2.a.b

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. 