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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 6045.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6045.g1 | 6045i3 | \([0, 1, 1, -195941, -33449224]\) | \(831958932702053269504/2361328125\) | \(2361328125\) | \([]\) | \(23328\) | \(1.4550\) | |
6045.g2 | 6045i2 | \([0, 1, 1, -2501, -43345]\) | \(1730766274822144/220896541125\) | \(220896541125\) | \([3]\) | \(7776\) | \(0.90572\) | |
6045.g3 | 6045i1 | \([0, 1, 1, -611, 5606]\) | \(25267247939584/39661245\) | \(39661245\) | \([3]\) | \(2592\) | \(0.35642\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6045.g have rank \(0\).
Complex multiplication
The elliptic curves in class 6045.g do not have complex multiplication.Modular form 6045.2.a.g
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.