↑

Properties

Label 20.0.27977981460...3313.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 17^{13}$
Root discriminant $16.69$
Ramified primes $7, 17$
Class number $1$
Class group Trivial
Galois group 20T144

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -70, 285, -547, 561, -429, 279, -80, 60, -229, 305, -303, 309, -262, 186, -131, 83, -42, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 18*x^18 - 42*x^17 + 83*x^16 - 131*x^15 + 186*x^14 - 262*x^13 + 309*x^12 - 303*x^11 + 305*x^10 - 229*x^9 + 60*x^8 - 80*x^7 + 279*x^6 - 429*x^5 + 561*x^4 - 547*x^3 + 285*x^2 - 70*x + 13)
 
gp: K = bnfinit(x^20 - 6*x^19 + 18*x^18 - 42*x^17 + 83*x^16 - 131*x^15 + 186*x^14 - 262*x^13 + 309*x^12 - 303*x^11 + 305*x^10 - 229*x^9 + 60*x^8 - 80*x^7 + 279*x^6 - 429*x^5 + 561*x^4 - 547*x^3 + 285*x^2 - 70*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 18 x^{18} - 42 x^{17} + 83 x^{16} - 131 x^{15} + 186 x^{14} - 262 x^{13} + 309 x^{12} - 303 x^{11} + 305 x^{10} - 229 x^{9} + 60 x^{8} - 80 x^{7} + 279 x^{6} - 429 x^{5} + 561 x^{4} - 547 x^{3} + 285 x^{2} - 70 x + 13 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2797798146085034747653313=7^{10}\cdot 17^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.69$
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:  $7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{29927521159527121} a^{19} + \frac{9569408838950432}{29927521159527121} a^{18} - \frac{8035302196872763}{29927521159527121} a^{17} - \frac{2505934021049129}{29927521159527121} a^{16} + \frac{5471018897402373}{29927521159527121} a^{15} - \frac{8340641894196674}{29927521159527121} a^{14} - \frac{12211123847774857}{29927521159527121} a^{13} - \frac{3326663644452}{2302117012271317} a^{12} + \frac{9476760816968767}{29927521159527121} a^{11} + \frac{6060245410243653}{29927521159527121} a^{10} - \frac{12213782011448156}{29927521159527121} a^{9} - \frac{6808006439530706}{29927521159527121} a^{8} - \frac{9760955652030723}{29927521159527121} a^{7} + \frac{10497213098859164}{29927521159527121} a^{6} - \frac{388653642127141}{2302117012271317} a^{5} - \frac{877929397574821}{2302117012271317} a^{4} - \frac{14127779568549385}{29927521159527121} a^{3} - \frac{6620367564325782}{29927521159527121} a^{2} + \frac{1895259408203103}{29927521159527121} a - \frac{846060990754277}{2302117012271317}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8656.62346224 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.3409076657.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ $20$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$