↑

Properties

Label 20.4.72379138322...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{58}\cdot 5^{14}\cdot 7^{6}\cdot 769^{4}$
Root discriminant $155.95$
Ramified primes $2, 5, 7, 769$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T872

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2941225, 0, -58320290, 0, -201345459, 0, -211359344, 0, -85646316, 0, -14722988, 0, -1118946, 0, -24096, 0, 1381, 0, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 78*x^18 + 1381*x^16 - 24096*x^14 - 1118946*x^12 - 14722988*x^10 - 85646316*x^8 - 211359344*x^6 - 201345459*x^4 - 58320290*x^2 + 2941225)
 
gp: K = bnfinit(x^20 + 78*x^18 + 1381*x^16 - 24096*x^14 - 1118946*x^12 - 14722988*x^10 - 85646316*x^8 - 211359344*x^6 - 201345459*x^4 - 58320290*x^2 + 2941225, 1)
 

Normalized defining polynomial

\( x^{20} + 78 x^{18} + 1381 x^{16} - 24096 x^{14} - 1118946 x^{12} - 14722988 x^{10} - 85646316 x^{8} - 211359344 x^{6} - 201345459 x^{4} - 58320290 x^{2} + 2941225 \)

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:  \(72379138322906453694958954086400000000000000=2^{58}\cdot 5^{14}\cdot 7^{6}\cdot 769^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.95$
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:  $2, 5, 7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{8} + \frac{2}{7} a^{6} - \frac{2}{7} a^{4} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} + \frac{2}{7} a^{7} - \frac{2}{7} a^{5} - \frac{3}{7} a^{3}$, $\frac{1}{147} a^{12} + \frac{1}{147} a^{10} - \frac{19}{147} a^{8} - \frac{44}{147} a^{6} + \frac{25}{147} a^{4} + \frac{4}{21} a^{2} - \frac{1}{3}$, $\frac{1}{147} a^{13} + \frac{1}{147} a^{11} - \frac{19}{147} a^{9} - \frac{44}{147} a^{7} + \frac{25}{147} a^{5} + \frac{4}{21} a^{3} - \frac{1}{3} a$, $\frac{1}{3087} a^{14} + \frac{8}{3087} a^{12} - \frac{53}{1029} a^{10} + \frac{13}{343} a^{8} - \frac{1312}{3087} a^{6} - \frac{181}{441} a^{4} + \frac{1}{21} a^{2} - \frac{1}{9}$, $\frac{1}{3087} a^{15} + \frac{8}{3087} a^{13} - \frac{53}{1029} a^{11} + \frac{13}{343} a^{9} - \frac{1312}{3087} a^{7} - \frac{181}{441} a^{5} + \frac{1}{21} a^{3} - \frac{1}{9} a$, $\frac{1}{64827} a^{16} - \frac{2}{21609} a^{14} + \frac{170}{64827} a^{12} - \frac{1130}{21609} a^{10} - \frac{20590}{64827} a^{8} - \frac{425}{1323} a^{6} + \frac{149}{1323} a^{4} + \frac{8}{27} a^{2} + \frac{11}{27}$, $\frac{1}{64827} a^{17} - \frac{2}{21609} a^{15} + \frac{170}{64827} a^{13} - \frac{1130}{21609} a^{11} - \frac{20590}{64827} a^{9} - \frac{425}{1323} a^{7} + \frac{149}{1323} a^{5} + \frac{8}{27} a^{3} + \frac{11}{27} a$, $\frac{1}{21507012260511248659074836355} a^{18} + \frac{709472312957634268778}{21507012260511248659074836355} a^{16} - \frac{2659374574200389634541069}{21507012260511248659074836355} a^{14} + \frac{57994260367212439829170414}{21507012260511248659074836355} a^{12} - \frac{954202579972799848961139901}{21507012260511248659074836355} a^{10} + \frac{352862252876898378657253061}{3072430322930178379867833765} a^{8} + \frac{14077140746617533010043923}{62702659651636293466690485} a^{6} + \frac{1002905330961463696880212}{2985840935792204450794785} a^{4} - \frac{848962723440209663346643}{2985840935792204450794785} a^{2} + \frac{78262189554389572386701}{255929223067903238639553}$, $\frac{1}{21507012260511248659074836355} a^{19} + \frac{709472312957634268778}{21507012260511248659074836355} a^{17} - \frac{2659374574200389634541069}{21507012260511248659074836355} a^{15} + \frac{57994260367212439829170414}{21507012260511248659074836355} a^{13} - \frac{954202579972799848961139901}{21507012260511248659074836355} a^{11} + \frac{352862252876898378657253061}{3072430322930178379867833765} a^{9} + \frac{14077140746617533010043923}{62702659651636293466690485} a^{7} + \frac{1002905330961463696880212}{2985840935792204450794785} a^{5} - \frac{848962723440209663346643}{2985840935792204450794785} a^{3} + \frac{78262189554389572386701}{255929223067903238639553} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 71377240237300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.4844429312000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
2Data not computed
5Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
769Data not computed