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Properties

Label 441525k
Number of curves $6$
Conductor $441525$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 441525k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441525.k5 441525k1 \([1, 1, 1, -16484038, 27573593906]\) \(-53297461115137/4513839183\) \(-41952137707687292859375\) \([2]\) \(41287680\) \(3.0859\) \(\Gamma_0(N)\)-optimal*
441525.k4 441525k2 \([1, 1, 1, -268889163, 1696981090656]\) \(231331938231569617/1472026689\) \(13681184433618971390625\) \([2, 2]\) \(82575360\) \(3.4325\) \(\Gamma_0(N)\)-optimal*
441525.k1 441525k3 \([1, 1, 1, -4302220038, 108612515925156]\) \(947531277805646290177/38367\) \(356587286825109375\) \([2]\) \(165150720\) \(3.7790\) \(\Gamma_0(N)\)-optimal*
441525.k3 441525k4 \([1, 1, 1, -274040288, 1628574150656]\) \(244883173420511137/18418027974129\) \(171179258841286154100140625\) \([2, 2]\) \(165150720\) \(3.7790\)  
441525.k6 441525k5 \([1, 1, 1, 262412587, 7228069259906]\) \(215015459663151503/2552757445339983\) \(-23725619708509447845992859375\) \([2]\) \(330301440\) \(4.1256\)  
441525.k2 441525k6 \([1, 1, 1, -892911163, -8348862096094]\) \(8471112631466271697/1662662681263647\) \(15452977152687609142370109375\) \([2]\) \(330301440\) \(4.1256\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 441525k1.

Rank

sage: E.rank()
 

The elliptic curves in class 441525k have rank \(1\).

Complex multiplication

The elliptic curves in class 441525k do not have complex multiplication.

Modular form 441525.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} + 2 q^{13} + q^{14} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.