Properties

Label 369600.hl
Number of curves $6$
Conductor $369600$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("hl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 369600.hl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.hl1 369600hl5 \([0, -1, 0, -4878720033, 131163302591937]\) \(3135316978843283198764801/571725\) \(2341785600000000\) \([2]\) \(94371840\) \(3.7436\)  
369600.hl2 369600hl3 \([0, -1, 0, -304920033, 2049502391937]\) \(765458482133960722801/326869475625\) \(1338857372160000000000\) \([2, 2]\) \(47185920\) \(3.3970\)  
369600.hl3 369600hl6 \([0, -1, 0, -303408033, 2070832175937]\) \(-754127868744065783521/15825714261328125\) \(-64822125614400000000000000\) \([2]\) \(94371840\) \(3.7436\)  
369600.hl4 369600hl4 \([0, -1, 0, -40712033, -52712856063]\) \(1821931919215868881/761147600816295\) \(3117660572943544320000000\) \([2]\) \(47185920\) \(3.3970\)  
369600.hl5 369600hl2 \([0, -1, 0, -19152033, 31694543937]\) \(189674274234120481/3859869269025\) \(15810024525926400000000\) \([2, 2]\) \(23592960\) \(3.0504\)  
369600.hl6 369600hl1 \([0, -1, 0, 55967, 1480359937]\) \(4733169839/231139696095\) \(-946748195205120000000\) \([2]\) \(11796480\) \(2.7039\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 369600.hl have rank \(1\).

Complex multiplication

The elliptic curves in class 369600.hl do not have complex multiplication.

Modular form 369600.2.a.hl

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} - q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.