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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4830b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4830.b3 | 4830b1 | \([1, 1, 0, -56723, -5223507]\) | \(20184279492242626489/11148103680\) | \(11148103680\) | \([2]\) | \(18432\) | \(1.2556\) | \(\Gamma_0(N)\)-optimal |
4830.b2 | 4830b2 | \([1, 1, 0, -57043, -5162003]\) | \(20527812941011798969/474091398849600\) | \(474091398849600\) | \([2, 2]\) | \(36864\) | \(1.6022\) | |
4830.b1 | 4830b3 | \([1, 1, 0, -125643, 9559557]\) | \(219353215817909485369/87028564162480920\) | \(87028564162480920\) | \([2]\) | \(73728\) | \(1.9487\) | |
4830.b4 | 4830b4 | \([1, 1, 0, 6437, -15940907]\) | \(29489595518609351/109830613939935000\) | \(-109830613939935000\) | \([2]\) | \(73728\) | \(1.9487\) |
Rank
sage: E.rank()
The elliptic curves in class 4830b have rank \(0\).
Complex multiplication
The elliptic curves in class 4830b do not have complex multiplication.Modular form 4830.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.