↑

Properties

Label 20.12.1017724340...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{75}\cdot 5^{20}\cdot 7^{10}$
Root discriminant $177.98$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T654

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5128, 64000, -827600, 1748160, 1348580, -1957440, -875040, 354560, 125080, 36480, 8048, 3680, 3780, -160, -400, -64, 90, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 90*x^16 - 64*x^15 - 400*x^14 - 160*x^13 + 3780*x^12 + 3680*x^11 + 8048*x^10 + 36480*x^9 + 125080*x^8 + 354560*x^7 - 875040*x^6 - 1957440*x^5 + 1348580*x^4 + 1748160*x^3 - 827600*x^2 + 64000*x + 5128)
 
gp: K = bnfinit(x^20 - 20*x^18 + 90*x^16 - 64*x^15 - 400*x^14 - 160*x^13 + 3780*x^12 + 3680*x^11 + 8048*x^10 + 36480*x^9 + 125080*x^8 + 354560*x^7 - 875040*x^6 - 1957440*x^5 + 1348580*x^4 + 1748160*x^3 - 827600*x^2 + 64000*x + 5128, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 90 x^{16} - 64 x^{15} - 400 x^{14} - 160 x^{13} + 3780 x^{12} + 3680 x^{11} + 8048 x^{10} + 36480 x^{9} + 125080 x^{8} + 354560 x^{7} - 875040 x^{6} - 1957440 x^{5} + 1348580 x^{4} + 1748160 x^{3} - 827600 x^{2} + 64000 x + 5128 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1017724340910230458282803200000000000000000000=2^{75}\cdot 5^{20}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.98$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{32} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{16} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{5}{16} a^{4} + \frac{1}{4} a^{2} + \frac{1}{8}$, $\frac{1}{32} a^{13} + \frac{1}{8} a^{11} - \frac{1}{16} a^{9} + \frac{1}{4} a^{7} + \frac{5}{16} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{10} + \frac{5}{16} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{11} + \frac{5}{16} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{4} a^{10} + \frac{3}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{32} a^{17} - \frac{1}{4} a^{11} + \frac{3}{16} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{64} a^{18} + \frac{3}{32} a^{10} - \frac{1}{4} a^{9} + \frac{3}{8} a^{6} + \frac{1}{4} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{52542816817164655271919218048693206421547497024} a^{19} + \frac{336947694821053381452923160326962744290790663}{52542816817164655271919218048693206421547497024} a^{18} + \frac{9163334558901520079555960390225272120251304}{820981512768197738623737782010831350336679641} a^{17} - \frac{306023571278946304237515379363074648168839371}{26271408408582327635959609024346603210773748512} a^{16} + \frac{162188942756412609296558336950047597561541175}{13135704204291163817979804512173301605386874256} a^{15} - \frac{406361586124763814083061873690392796385373955}{26271408408582327635959609024346603210773748512} a^{14} - \frac{184234782782131771109351223043169830847465363}{26271408408582327635959609024346603210773748512} a^{13} - \frac{62958316810428816363234252175318351643340625}{26271408408582327635959609024346603210773748512} a^{12} + \frac{5770378839069418347190390683913492320882887443}{26271408408582327635959609024346603210773748512} a^{11} - \frac{501624206728265747336177922640154034682355945}{26271408408582327635959609024346603210773748512} a^{10} + \frac{2144324608916874515278215086218370273772237127}{13135704204291163817979804512173301605386874256} a^{9} - \frac{101548846555118661686912178712767545469057509}{1641963025536395477247475564021662700673359282} a^{8} + \frac{71323940036199856850594916722715578565907612}{820981512768197738623737782010831350336679641} a^{7} - \frac{2000750592996812274685053537298989262834416193}{13135704204291163817979804512173301605386874256} a^{6} - \frac{2532827169247643111331501502692901909186537731}{13135704204291163817979804512173301605386874256} a^{5} - \frac{115675745421381544838281335642579086067475509}{13135704204291163817979804512173301605386874256} a^{4} - \frac{1784231955992260774525298566584718777790622367}{6567852102145581908989902256086650802693437128} a^{3} + \frac{1332448507448429771595462587213785960034780989}{3283926051072790954494951128043325401346718564} a^{2} + \frac{950752893878684947925435748831754398450341161}{6567852102145581908989902256086650802693437128} a - \frac{2294650353112842285588988237771995258218559167}{6567852102145581908989902256086650802693437128}$

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:  $15$
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:  \( 6087162868560000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 10.6.1409873346560000000000.1

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 24 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $20$ $20$

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$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$