Properties

Label 28224.dw
Number of curves $4$
Conductor $28224$
CM \(\Q(\sqrt{-7}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.dw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.dw1 28224bn4 \([0, 0, 0, -1049580, -413855568]\) \(16581375\) \(7711694390034432\) \([2]\) \(229376\) \(2.1094\)   \(-28\)
28224.dw2 28224bn3 \([0, 0, 0, -61740, -7260624]\) \(-3375\) \(-7711694390034432\) \([2]\) \(114688\) \(1.7628\)   \(-7\)
28224.dw3 28224bn2 \([0, 0, 0, -21420, 1206576]\) \(16581375\) \(65548320768\) \([2]\) \(32768\) \(1.1365\)   \(-28\)
28224.dw4 28224bn1 \([0, 0, 0, -1260, 21168]\) \(-3375\) \(-65548320768\) \([2]\) \(16384\) \(0.78989\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

sage: E.rank()
 

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

Complex multiplication

Each elliptic curve in class 28224.dw has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 28224.2.a.dw

sage: E.q_eigenform(10)
 
\(q + 4q^{11} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.