Properties

Label 377520.ha
Number of curves $6$
Conductor $377520$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ha1")
 
E.isogeny_class()
 

Elliptic curves in class 377520.ha

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
377520.ha1 377520ha6 \([0, 1, 0, -17453080, -28070245612]\) \(81025909800741361/11088090\) \(80458661103575040\) \([2]\) \(15728640\) \(2.6562\)  
377520.ha2 377520ha3 \([0, 1, 0, -1635960, 804210900]\) \(66730743078481/60937500\) \(442181625600000000\) \([2]\) \(7864320\) \(2.3096\)  
377520.ha3 377520ha4 \([0, 1, 0, -1093880, -436284972]\) \(19948814692561/231344100\) \(1678705398333849600\) \([2, 2]\) \(7864320\) \(2.3096\)  
377520.ha4 377520ha5 \([0, 1, 0, -222680, -1111290732]\) \(-168288035761/73415764890\) \(-532727832020145315840\) \([2]\) \(15728640\) \(2.6562\)  
377520.ha5 377520ha2 \([0, 1, 0, -125880, 6284628]\) \(30400540561/15210000\) \(110368533749760000\) \([2, 2]\) \(3932160\) \(1.9630\)  
377520.ha6 377520ha1 \([0, 1, 0, 29000, 770900]\) \(371694959/249600\) \(-1811175938457600\) \([2]\) \(1966080\) \(1.6164\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 377520.ha have rank \(1\).

Complex multiplication

The elliptic curves in class 377520.ha do not have complex multiplication.

Modular form 377520.2.a.ha

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.