Properties

Label 1568.d
Number of curves $2$
Conductor $1568$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 1568.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1568.d1 1568b2 \([0, 0, 0, -1372, 0]\) \(1728\) \(165288374272\) \([2]\) \(1792\) \(0.84205\)   \(-4\)
1568.d2 1568b1 \([0, 0, 0, 343, 0]\) \(1728\) \(-2582630848\) \([2]\) \(896\) \(0.49547\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 1568.d have rank \(0\).

Complex multiplication

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

Modular form 1568.2.a.d

sage: E.q_eigenform(10)
 
\(q - 4 q^{5} - 3 q^{9} + 4 q^{13} - 8 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.