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SageMath
E = EllipticCurve("fm1")
E.isogeny_class()
Elliptic curves in class 129360.fm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.fm1 | 129360gr1 | \([0, 1, 0, -110021, 29023455]\) | \(-4890195460096/9282994875\) | \(-279586576396512000\) | \([]\) | \(1492992\) | \(2.0375\) | \(\Gamma_0(N)\)-optimal |
129360.fm2 | 129360gr2 | \([0, 1, 0, 948379, -615436305]\) | \(3132137615458304/7250937873795\) | \(-218384791018011636480\) | \([]\) | \(4478976\) | \(2.5868\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.fm have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.fm do not have complex multiplication.Modular form 129360.2.a.fm
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.