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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1001.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1001.b1 | 1001b3 | \([1, -1, 1, -9916, -377564]\) | \(107818231938348177/4463459\) | \(4463459\) | \([2]\) | \(608\) | \(0.76223\) | |
1001.b2 | 1001b4 | \([1, -1, 1, -1006, 2552]\) | \(112489728522417/62811265517\) | \(62811265517\) | \([2]\) | \(608\) | \(0.76223\) | |
1001.b3 | 1001b2 | \([1, -1, 1, -621, -5764]\) | \(26444947540257/169338169\) | \(169338169\) | \([2, 2]\) | \(304\) | \(0.41565\) | |
1001.b4 | 1001b1 | \([1, -1, 1, -16, -198]\) | \(-426957777/17320303\) | \(-17320303\) | \([4]\) | \(152\) | \(0.069081\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1001.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1001.b do not have complex multiplication.Modular form 1001.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.