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Properties

Label 12274e
Number of curves $2$
Conductor $12274$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 12274e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12274.h2 12274e1 \([1, 1, 0, -87008, -9865780]\) \(1548415333009/8861828\) \(416912505530468\) \([2]\) \(161280\) \(1.6467\) \(\Gamma_0(N)\)-optimal
12274.h1 12274e2 \([1, 1, 0, -1390218, -631496950]\) \(6316133726112049/208658\) \(9816499437698\) \([2]\) \(322560\) \(1.9933\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12274e have rank \(0\).

Complex multiplication

The elliptic curves in class 12274e do not have complex multiplication.

Modular form 12274.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} - q^{8} + q^{9} - 4 q^{10} + 2 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} + 8 q^{15} + q^{16} - q^{17} - q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.