Properties

Label 485100.he
Number of curves $4$
Conductor $485100$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("he1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 485100.he

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.he1 485100he4 \([0, 0, 0, -546799575, 2594830414750]\) \(52702650535889104/22020583921875\) \(7554480260536743187500000000\) \([2]\) \(286654464\) \(4.0464\)  
485100.he2 485100he2 \([0, 0, 0, -471388575, 3939277567750]\) \(33766427105425744/9823275\) \(3370016769065100000000\) \([2]\) \(95551488\) \(3.4971\) \(\Gamma_0(N)\)-optimal*
485100.he3 485100he1 \([0, 0, 0, -29341200, 62080041625]\) \(-130287139815424/2250652635\) \(-48257436555594708750000\) \([2]\) \(47775744\) \(3.1505\) \(\Gamma_0(N)\)-optimal*
485100.he4 485100he3 \([0, 0, 0, 113542800, 297499292125]\) \(7549996227362816/6152409907875\) \(-131917083150101525718750000\) \([2]\) \(143327232\) \(3.6998\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 485100.he1.

Rank

sage: E.rank()
 

The elliptic curves in class 485100.he have rank \(1\).

Complex multiplication

The elliptic curves in class 485100.he do not have complex multiplication.

Modular form 485100.2.a.he

sage: E.q_eigenform(10)
 
\(q + q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.