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Properties

Label 20.4.29521807615...8224.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 89^{4}$
Root discriminant $23.63$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23, -351, 1552, -2463, 620, 945, 19, 1032, -1046, -380, 51, -96, 251, 45, 6, 16, -25, -2, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 2*x^17 - 25*x^16 + 16*x^15 + 6*x^14 + 45*x^13 + 251*x^12 - 96*x^11 + 51*x^10 - 380*x^9 - 1046*x^8 + 1032*x^7 + 19*x^6 + 945*x^5 + 620*x^4 - 2463*x^3 + 1552*x^2 - 351*x + 23)
 
gp: K = bnfinit(x^20 + x^18 - 2*x^17 - 25*x^16 + 16*x^15 + 6*x^14 + 45*x^13 + 251*x^12 - 96*x^11 + 51*x^10 - 380*x^9 - 1046*x^8 + 1032*x^7 + 19*x^6 + 945*x^5 + 620*x^4 - 2463*x^3 + 1552*x^2 - 351*x + 23, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 2 x^{17} - 25 x^{16} + 16 x^{15} + 6 x^{14} + 45 x^{13} + 251 x^{12} - 96 x^{11} + 51 x^{10} - 380 x^{9} - 1046 x^{8} + 1032 x^{7} + 19 x^{6} + 945 x^{5} + 620 x^{4} - 2463 x^{3} + 1552 x^{2} - 351 x + 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2952180761584785045865268224=2^{10}\cdot 11^{16}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.63$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{909355344121579593759467356961} a^{19} - \frac{354724149220065442715660218040}{909355344121579593759467356961} a^{18} - \frac{420043397379273532621742509587}{909355344121579593759467356961} a^{17} + \frac{206104095212943189329115442034}{909355344121579593759467356961} a^{16} - \frac{334994384082316279514389679201}{909355344121579593759467356961} a^{15} + \frac{93008305134172013051207061077}{909355344121579593759467356961} a^{14} - \frac{246981161135861402262407823644}{909355344121579593759467356961} a^{13} + \frac{237737081007605321187509902138}{909355344121579593759467356961} a^{12} - \frac{284633879250306249034736116797}{909355344121579593759467356961} a^{11} + \frac{287466308481899012192627400861}{909355344121579593759467356961} a^{10} - \frac{295048280327155690016347747038}{909355344121579593759467356961} a^{9} + \frac{106348201987467811969577481599}{909355344121579593759467356961} a^{8} + \frac{314444478389632996348855414807}{909355344121579593759467356961} a^{7} - \frac{409488619145398136252846676782}{909355344121579593759467356961} a^{6} + \frac{422666409974784221056451244858}{909355344121579593759467356961} a^{5} + \frac{359101021712747670433354078611}{909355344121579593759467356961} a^{4} - \frac{293534826283702851329921415106}{909355344121579593759467356961} a^{3} - \frac{318055233371966822056389919227}{909355344121579593759467356961} a^{2} + \frac{242173675007374849284954234033}{909355344121579593759467356961} a - \frac{419521164100690263266244611067}{909355344121579593759467356961}$

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:  $11$
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:  \( 445493.735906 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
89Data not computed