Properties

Label 331200w
Number of curves $4$
Conductor $331200$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 331200w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
331200.w4 331200w1 \([0, 0, 0, 129300, 1294000]\) \(80062991/46575\) \(-139072204800000000\) \([2]\) \(3932160\) \(1.9789\) \(\Gamma_0(N)\)-optimal
331200.w3 331200w2 \([0, 0, 0, -518700, 10366000]\) \(5168743489/2975625\) \(8885168640000000000\) \([2, 2]\) \(7864320\) \(2.3255\)  
331200.w1 331200w3 \([0, 0, 0, -5918700, 5529166000]\) \(7679186557489/20988075\) \(62670056140800000000\) \([2]\) \(15728640\) \(2.6721\)  
331200.w2 331200w4 \([0, 0, 0, -5486700, -4927826000]\) \(6117442271569/26953125\) \(80481600000000000000\) \([2]\) \(15728640\) \(2.6721\)  

Rank

sage: E.rank()
 

The elliptic curves in class 331200w have rank \(0\).

Complex multiplication

The elliptic curves in class 331200w do not have complex multiplication.

Modular form 331200.2.a.w

sage: E.q_eigenform(10)
 
\(q - 4 q^{7} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.