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SageMath
E = EllipticCurve("m1")
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)
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.