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Properties

Label 458640hd
Number of curves $4$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640hd have rank \(1\).

Complex multiplication

The elliptic curves in class 458640hd do not have complex multiplication.

Modular form 458640.2.a.hd

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + 6 q^{11} - q^{13} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 458640hd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.hd4 458640hd1 \([0, 0, 0, -2753163, 1482491962]\) \(177381177331203/29679104000\) \(386155956123205632000\) \([2]\) \(15925248\) \(2.6713\) \(\Gamma_0(N)\)-optimal
458640.hd3 458640hd2 \([0, 0, 0, -12537483, -15681162182]\) \(16751080718799363/1529437000000\) \(19899563243728896000000\) \([2]\) \(31850496\) \(3.0179\)  
458640.hd2 458640hd3 \([0, 0, 0, -62023563, -187853068998]\) \(2781982314427707/2703013040\) \(25638205330378267361280\) \([2]\) \(47775744\) \(3.2206\)  
458640.hd1 458640hd4 \([0, 0, 0, -992145483, -12028491134982]\) \(11387025941627437947/10765300\) \(102109374893404569600\) \([2]\) \(95551488\) \(3.5672\)