Properties

Label 1728.m
Number of curves $2$
Conductor $1728$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1728.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1728.m1 1728m1 \([0, 0, 0, 0, -32]\) \(0\) \(-442368\) \([]\) \(192\) \(-0.23779\) \(\Gamma_0(N)\)-optimal \(-3\)
1728.m2 1728m2 \([0, 0, 0, 0, 864]\) \(0\) \(-322486272\) \([]\) \(576\) \(0.31151\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 1728.m have rank \(0\).

Complex multiplication

Each elliptic curve in class 1728.m has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1728.2.a.m

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.