Properties

Label 12342.w
Number of curves $4$
Conductor $12342$
CM no
Rank $0$
Graph

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

Elliptic curves in class 12342.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12342.w1 12342v3 \([1, 1, 1, -58556682, -172490345301]\) \(12534210458299016895673/315581882565708\) \(559072555459988230188\) \([2]\) \(1843200\) \(3.0892\)  
12342.w2 12342v2 \([1, 1, 1, -3799342, -2479756069]\) \(3423676911662954233/483711578981136\) \(856924568571400273296\) \([2, 2]\) \(921600\) \(2.7426\)  
12342.w3 12342v1 \([1, 1, 1, -1001822, 346858139]\) \(62768149033310713/6915442583808\) \(12251128379213484288\) \([4]\) \(460800\) \(2.3961\) \(\Gamma_0(N)\)-optimal
12342.w4 12342v4 \([1, 1, 1, 6197678, -13316525749]\) \(14861225463775641287/51859390496937804\) \(-91872073688145632992044\) \([2]\) \(1843200\) \(3.0892\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12342.w have rank \(0\).

Complex multiplication

The elliptic curves in class 12342.w do not have complex multiplication.

Modular form 12342.2.a.w

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