Properties

Label 270480.hk
Number of curves $4$
Conductor $270480$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hk1")
 
E.isogeny_class()
 

Elliptic curves in class 270480.hk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
270480.hk1 270480hk3 \([0, 1, 0, -1445320, 623962100]\) \(692895692874169/51420783750\) \(24779177113205760000\) \([2]\) \(7077888\) \(2.4671\)  
270480.hk2 270480hk2 \([0, 1, 0, -292840, -49547212]\) \(5763259856089/1143116100\) \(550856564936294400\) \([2, 2]\) \(3538944\) \(2.1205\)  
270480.hk3 270480hk1 \([0, 1, 0, -277160, -56251980]\) \(4886171981209/270480\) \(130341689425920\) \([2]\) \(1769472\) \(1.7740\) \(\Gamma_0(N)\)-optimal
270480.hk4 270480hk4 \([0, 1, 0, 608760, -293700492]\) \(51774168853511/107398242630\) \(-51754171790036459520\) \([2]\) \(7077888\) \(2.4671\)  

Rank

sage: E.rank()
 

The elliptic curves in class 270480.hk have rank \(1\).

Complex multiplication

The elliptic curves in class 270480.hk do not have complex multiplication.

Modular form 270480.2.a.hk

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} - 4 q^{11} - 2 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.