Properties

Label 1568.e
Number of curves $4$
Conductor $1568$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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

Elliptic curves in class 1568.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1568.e1 1568g2 \([0, 0, 0, -539, -4802]\) \(287496\) \(60236288\) \([2]\) \(384\) \(0.35557\)   \(-16\)
1568.e2 1568g3 \([0, 0, 0, -539, 4802]\) \(287496\) \(60236288\) \([2]\) \(384\) \(0.35557\)   \(-16\)
1568.e3 1568g1 \([0, 0, 0, -49, 0]\) \(1728\) \(7529536\) \([2, 2]\) \(192\) \(0.0089957\) \(\Gamma_0(N)\)-optimal \(-4\)
1568.e4 1568g4 \([0, 0, 0, 196, 0]\) \(1728\) \(-481890304\) \([2]\) \(384\) \(0.35557\)   \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 1568.e have rank \(1\).

Complex multiplication

Each elliptic curve in class 1568.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 1568.2.a.e

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20})\)  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.