Properties

Label 144.a
Number of curves $4$
Conductor $144$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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

Elliptic curves in class 144.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
144.a1 144a4 \([0, 0, 0, -135, 594]\) \(54000\) \(5038848\) \([2]\) \(24\) \(0.080464\)   \(-12\)
144.a2 144a2 \([0, 0, 0, -15, -22]\) \(54000\) \(6912\) \([2]\) \(8\) \(-0.46884\)   \(-12\)
144.a3 144a1 \([0, 0, 0, 0, -1]\) \(0\) \(-432\) \([2]\) \(4\) \(-0.81542\) \(\Gamma_0(N)\)-optimal \(-3\)
144.a4 144a3 \([0, 0, 0, 0, 27]\) \(0\) \(-314928\) \([2]\) \(12\) \(-0.26611\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 144.a have rank \(0\).

Complex multiplication

Each elliptic curve in class 144.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 144.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.