Properties

Label 26912a
Number of curves $4$
Conductor $26912$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 26912a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
26912.d3 26912a1 \([0, 0, 0, -841, 0]\) \(1728\) \(38068692544\) \([2, 2]\) \(11200\) \(0.71969\) \(\Gamma_0(N)\)-optimal \(-4\)
26912.d4 26912a2 \([0, 0, 0, 3364, 0]\) \(1728\) \(-2436396322816\) \([2]\) \(22400\) \(1.0663\)   \(-4\)
26912.d1 26912a3 \([0, 0, 0, -9251, -341446]\) \(287496\) \(304549540352\) \([2]\) \(22400\) \(1.0663\)   \(-16\)
26912.d2 26912a4 \([0, 0, 0, -9251, 341446]\) \(287496\) \(304549540352\) \([2]\) \(22400\) \(1.0663\)   \(-16\)

Rank

sage: E.rank()
 

The elliptic curves in class 26912a have rank \(1\).

Complex multiplication

Each elliptic curve in class 26912a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 26912.2.a.a

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