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Properties

Label 2730e
Number of curves $2$
Conductor $2730$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 2730e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2730.e1 2730e1 \([1, 1, 0, -75183, 7787493]\) \(46999332667159819129/788827220213760\) \(788827220213760\) \([2]\) \(21120\) \(1.6566\) \(\Gamma_0(N)\)-optimal
2730.e2 2730e2 \([1, 1, 0, -3503, 22109157]\) \(-4755955967570809/211193136454809600\) \(-211193136454809600\) \([2]\) \(42240\) \(2.0032\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2730e have rank \(1\).

Complex multiplication

The elliptic curves in class 2730e do not have complex multiplication.

Modular form 2730.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + 2 q^{11} - q^{12} + q^{13} - q^{14} + q^{15} + q^{16} + 4 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.