Properties

Label 3025.d
Number of curves $2$
Conductor $3025$
CM \(\Q(\sqrt{-11}) \)
Rank $1$
Graph

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

Elliptic curves in class 3025.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
3025.d1 3025a2 \([0, 1, 1, -22183, -1312206]\) \(-32768\) \(-36842932671875\) \([]\) \(4752\) \(1.3806\)   \(-11\)
3025.d2 3025a1 \([0, 1, 1, -183, 919]\) \(-32768\) \(-20796875\) \([]\) \(432\) \(0.18165\) \(\Gamma_0(N)\)-optimal \(-11\)

Rank

sage: E.rank()
 

The elliptic curves in class 3025.d have rank \(1\).

Complex multiplication

Each elliptic curve in class 3025.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).

Modular form 3025.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{4} - 2q^{9} - 2q^{12} + 4q^{16} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.