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Properties

Label 112530.w
Number of curves $4$
Conductor $112530$
CM no
Rank $2$
Graph

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

Elliptic curves in class 112530.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112530.w1 112530bb4 \([1, 0, 1, -76891994, 259512961376]\) \(28379906689597370652529/1357352437500\) \(2404632641529937500\) \([2]\) \(12441600\) \(3.0043\)  
112530.w2 112530bb3 \([1, 0, 1, -4797774, 4068721072]\) \(-6894246873502147249/47925198774000\) \(-84902413065266214000\) \([2]\) \(6220800\) \(2.6577\)  
112530.w3 112530bb2 \([1, 0, 1, -1032254, 289997552]\) \(68663623745397169/19216056254400\) \(34042415834101118400\) \([2]\) \(4147200\) \(2.4550\)  
112530.w4 112530bb1 \([1, 0, 1, 168066, 29768176]\) \(296354077829711/387386634240\) \(-686279053140848640\) \([2]\) \(2073600\) \(2.1084\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 112530.w have rank \(2\).

Complex multiplication

The elliptic curves in class 112530.w do not have complex multiplication.

Modular form 112530.2.a.w

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 4 q^{13} + 2 q^{14} - q^{15} + q^{16} - 6 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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 LMFDB 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

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

The vertices are labelled with LMFDB labels.