Properties

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

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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.

sage: E = EllipticCurve("b1")
 
sage: 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 form 147.2.a.b

sage: E.q_eigenform(10)
 
\(q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + q^{9} + 4q^{10} - 2q^{11} - 2q^{12} - q^{13} - 2q^{15} - 4q^{16} + 2q^{18} - q^{19} + O(q^{20})\)  Toggle raw display

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.