Properties

Label 333270.bp
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bp1 333270bp1 \([1, -1, 0, -674574, 209365380]\) \(8493409990827/185150000\) \(740038810905450000\) \([2]\) \(5406720\) \(2.2171\) \(\Gamma_0(N)\)-optimal
333270.bp2 333270bp2 \([1, -1, 0, 55446, 638179128]\) \(4716275733/44023437500\) \(-175960315093007812500\) \([2]\) \(10813440\) \(2.5636\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.bp have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.bp do not have complex multiplication.

Modular form 333270.2.a.bp

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} - 2q^{11} + 2q^{13} + q^{14} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.