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Properties

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

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Show commands: SageMath
E = EllipticCurve("bt1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bt1 333270bt1 \([1, -1, 0, -23455550634, -578091962174860]\) \(489781415227546051766883/233890092903563264000\) \(681506708102918490519521722368000\) \([2]\) \(1634992128\) \(4.9940\) \(\Gamma_0(N)\)-optimal
333270.bt2 333270bt2 \([1, -1, 0, 84190278486, -4396999983533452]\) \(22649115256119592694355357/15973509811739648000000\) \(-46543459594660369411793458176000000\) \([2]\) \(3269984256\) \(5.3406\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.bt

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + 2 q^{11} - 6 q^{13} + q^{14} + q^{16} - 6 q^{17} + O(q^{20})\) Copy content 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.