Properties

Label 972.a
Number of curves $2$
Conductor $972$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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

Elliptic curves in class 972.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
972.a1 972d2 \([0, 0, 0, 0, -972]\) \(0\) \(-408146688\) \([]\) \(324\) \(0.33114\)   \(-3\)
972.a2 972d1 \([0, 0, 0, 0, 36]\) \(0\) \(-559872\) \([3]\) \(108\) \(-0.21816\) \(\Gamma_0(N)\)-optimal \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 972.a have rank \(1\).

Complex multiplication

Each elliptic curve in class 972.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 972.2.a.a

sage: E.q_eigenform(10)
 
\(q - 4 q^{7} + 5 q^{13} - 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.