Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1568g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
1568.e3 1568g1 [0, 0, 0, -49, 0] [2, 2] 192 \(\Gamma_0(N)\)-optimal -4
1568.e1 1568g2 [0, 0, 0, -539, -4802] [2] 384   -16
1568.e2 1568g3 [0, 0, 0, -539, 4802] [2] 384   -16
1568.e4 1568g4 [0, 0, 0, 196, 0] [2] 384   -4

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1568.2.a.g

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.