Properties

Label 576.f
Number of curves $4$
Conductor $576$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 576.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
576.f1 576e4 \([0, 0, 0, -540, 4752]\) \(54000\) \(322486272\) \([2]\) \(192\) \(0.42704\)   \(-12\)
576.f2 576e2 \([0, 0, 0, -60, -176]\) \(54000\) \(442368\) \([2]\) \(64\) \(-0.12227\)   \(-12\)
576.f3 576e1 \([0, 0, 0, 0, -8]\) \(0\) \(-27648\) \([2]\) \(32\) \(-0.46884\) \(\Gamma_0(N)\)-optimal \(-3\)
576.f4 576e3 \([0, 0, 0, 0, 216]\) \(0\) \(-20155392\) \([2]\) \(96\) \(0.080464\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 576.f have rank \(0\).

Complex multiplication

Each elliptic curve in class 576.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 576.2.a.f

sage: E.q_eigenform(10)
 
\(q + 4 q^{7} - 2 q^{13} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.