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Properties

Label 20.20.1245907905...1889.1
Degree $20$
Signature $[20, 0]$
Discriminant $17^{2}\cdot 401^{11}$
Root discriminant $35.87$
Ramified primes $17, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -15, 29, 331, -1061, -1592, 8376, -2769, -17904, 15564, 12766, -17991, -955, 7492, -1418, -1294, 396, 89, -35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 35*x^18 + 89*x^17 + 396*x^16 - 1294*x^15 - 1418*x^14 + 7492*x^13 - 955*x^12 - 17991*x^11 + 12766*x^10 + 15564*x^9 - 17904*x^8 - 2769*x^7 + 8376*x^6 - 1592*x^5 - 1061*x^4 + 331*x^3 + 29*x^2 - 15*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 35*x^18 + 89*x^17 + 396*x^16 - 1294*x^15 - 1418*x^14 + 7492*x^13 - 955*x^12 - 17991*x^11 + 12766*x^10 + 15564*x^9 - 17904*x^8 - 2769*x^7 + 8376*x^6 - 1592*x^5 - 1061*x^4 + 331*x^3 + 29*x^2 - 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 35 x^{18} + 89 x^{17} + 396 x^{16} - 1294 x^{15} - 1418 x^{14} + 7492 x^{13} - 955 x^{12} - 17991 x^{11} + 12766 x^{10} + 15564 x^{9} - 17904 x^{8} - 2769 x^{7} + 8376 x^{6} - 1592 x^{5} - 1061 x^{4} + 331 x^{3} + 29 x^{2} - 15 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12459079056924067396846384471889=17^{2}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.87$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $17, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{2}{17} a^{17} + \frac{3}{17} a^{16} + \frac{8}{17} a^{15} + \frac{8}{17} a^{14} - \frac{2}{17} a^{13} + \frac{4}{17} a^{12} + \frac{2}{17} a^{11} + \frac{6}{17} a^{10} - \frac{6}{17} a^{9} + \frac{2}{17} a^{8} + \frac{7}{17} a^{7} + \frac{8}{17} a^{5} - \frac{7}{17} a^{4} - \frac{3}{17} a^{3} - \frac{8}{17} a^{2} + \frac{5}{17} a - \frac{4}{17}$, $\frac{1}{735132118601453675049} a^{19} + \frac{3943885781110526381}{735132118601453675049} a^{18} + \frac{14347105812413346361}{81681346511272630561} a^{17} - \frac{240449844109693402162}{735132118601453675049} a^{16} + \frac{204782568267019710344}{735132118601453675049} a^{15} + \frac{30080197063209381727}{735132118601453675049} a^{14} + \frac{101448165390472880048}{735132118601453675049} a^{13} - \frac{11301378360529296587}{245044039533817891683} a^{12} - \frac{38354197167946928392}{735132118601453675049} a^{11} - \frac{24790097709431628220}{735132118601453675049} a^{10} + \frac{1618047840274194156}{4804785088898390033} a^{9} - \frac{105380168676099980495}{245044039533817891683} a^{8} - \frac{30601422372767673904}{81681346511272630561} a^{7} - \frac{85075132070009888534}{245044039533817891683} a^{6} - \frac{40350859924780526448}{81681346511272630561} a^{5} - \frac{66009269677217267150}{735132118601453675049} a^{4} + \frac{197833848752676305552}{735132118601453675049} a^{3} + \frac{19351478305395025044}{81681346511272630561} a^{2} - \frac{335355847071048815200}{735132118601453675049} a + \frac{259685339043716982830}{735132118601453675049}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1370360256.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed