Properties

Label 28224.di
Number of curves $4$
Conductor $28224$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("di1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.di

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.di1 28224di4 \([0, 0, 0, -26460, -1629936]\) \(54000\) \(37940187414528\) \([2]\) \(55296\) \(1.4000\)   \(-12\)
28224.di2 28224di2 \([0, 0, 0, -2940, 60368]\) \(54000\) \(52044152832\) \([2]\) \(18432\) \(0.85069\)   \(-12\)
28224.di3 28224di3 \([0, 0, 0, 0, -74088]\) \(0\) \(-2371261713408\) \([2]\) \(27648\) \(1.0534\)   \(-3\)
28224.di4 28224di1 \([0, 0, 0, 0, 2744]\) \(0\) \(-3252759552\) \([2]\) \(9216\) \(0.50411\) \(\Gamma_0(N)\)-optimal \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 28224.di have rank \(1\).

Complex multiplication

Each elliptic curve in class 28224.di has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 28224.2.a.di

sage: E.q_eigenform(10)
 
\(q + 2q^{13} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.