Properties

Label 453152.u
Number of curves $4$
Conductor $453152$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 453152.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
453152.u1 453152u4 \([0, 0, 0, -155771, -23592226]\) \(287496\) \(1453957557903872\) \([2]\) \(1966080\) \(1.7722\)   \(-16\)
453152.u2 453152u2 \([0, 0, 0, -155771, 23592226]\) \(287496\) \(1453957557903872\) \([2]\) \(1966080\) \(1.7722\) \(\Gamma_0(N)\)-optimal* \(-16\)
453152.u3 453152u1 \([0, 0, 0, -14161, 0]\) \(1728\) \(181744694737984\) \([2, 2]\) \(983040\) \(1.4256\) \(\Gamma_0(N)\)-optimal* \(-4\)
453152.u4 453152u3 \([0, 0, 0, 56644, 0]\) \(1728\) \(-11631660463230976\) \([2]\) \(1966080\) \(1.7722\)   \(-4\)
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 453152.u1.

Rank

sage: E.rank()
 

The elliptic curves in class 453152.u have rank \(1\).

Complex multiplication

Each elliptic curve in class 453152.u has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 453152.2.a.u

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - 3 q^{9} - 6 q^{13} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.