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Properties

Label 20.4.18512297918...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $57.99$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1125766475, 25546650, -71093575, 116976750, -127061025, 93177780, -49412490, 16135260, -4537845, 335370, -67359, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 67359*x^10 + 335370*x^9 - 4537845*x^8 + 16135260*x^7 - 49412490*x^6 + 93177780*x^5 - 127061025*x^4 + 116976750*x^3 - 71093575*x^2 + 25546650*x + 1125766475)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 67359*x^10 + 335370*x^9 - 4537845*x^8 + 16135260*x^7 - 49412490*x^6 + 93177780*x^5 - 127061025*x^4 + 116976750*x^3 - 71093575*x^2 + 25546650*x + 1125766475, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} - 67359 x^{10} + 335370 x^{9} - 4537845 x^{8} + 16135260 x^{7} - 49412490 x^{6} + 93177780 x^{5} - 127061025 x^{4} + 116976750 x^{3} - 71093575 x^{2} + 25546650 x + 1125766475 \)

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:  \(185122979184640000000000000000000000=2^{38}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.99$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{5} + \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{8} + \frac{3}{49} a^{7} + \frac{2}{49} a^{6} + \frac{1}{49} a^{5} + \frac{2}{49} a^{4} + \frac{20}{49} a^{3} + \frac{9}{49} a^{2} + \frac{11}{49} a + \frac{15}{49}$, $\frac{1}{49} a^{9} + \frac{2}{49} a^{6} - \frac{1}{49} a^{5} + \frac{12}{49} a^{3} + \frac{12}{49} a^{2} + \frac{17}{49} a - \frac{3}{49}$, $\frac{1}{245} a^{10} - \frac{1}{49} a^{7} - \frac{3}{49} a^{6} + \frac{2}{35} a^{5} + \frac{1}{49} a^{4} + \frac{8}{49} a^{3} + \frac{2}{49} a^{2} - \frac{9}{49} a - \frac{2}{7}$, $\frac{1}{245} a^{11} - \frac{11}{245} a^{6} + \frac{2}{49} a^{5} + \frac{3}{49} a^{4} + \frac{8}{49} a^{3} + \frac{1}{7} a^{2} - \frac{3}{49} a + \frac{22}{49}$, $\frac{1}{1715} a^{12} + \frac{1}{1715} a^{11} + \frac{2}{1715} a^{10} + \frac{2}{343} a^{9} + \frac{1}{343} a^{8} - \frac{111}{1715} a^{7} - \frac{106}{1715} a^{6} + \frac{13}{1715} a^{5} + \frac{22}{343} a^{4} - \frac{107}{343} a^{3} + \frac{139}{343} a^{2} - \frac{115}{343} a + \frac{171}{343}$, $\frac{1}{1715} a^{13} + \frac{1}{1715} a^{11} + \frac{1}{1715} a^{10} - \frac{1}{343} a^{9} - \frac{11}{1715} a^{8} + \frac{22}{343} a^{7} - \frac{8}{245} a^{6} + \frac{104}{1715} a^{5} + \frac{4}{343} a^{4} - \frac{27}{343} a^{3} + \frac{68}{343} a^{2} + \frac{90}{343} a + \frac{46}{343}$, $\frac{1}{1715} a^{14} + \frac{2}{245} a^{9} - \frac{10}{343} a^{7} - \frac{1}{49} a^{6} - \frac{1}{49} a^{5} + \frac{2}{49} a^{4} + \frac{13}{49} a^{3} - \frac{20}{49} a^{2} + \frac{5}{49} a - \frac{17}{343}$, $\frac{1}{1715} a^{15} - \frac{3}{343} a^{8} - \frac{3}{49} a^{7} - \frac{13}{245} a^{5} - \frac{1}{49} a^{4} - \frac{23}{49} a^{3} + \frac{10}{49} a^{2} - \frac{108}{343} a + \frac{22}{49}$, $\frac{1}{60025} a^{16} - \frac{8}{60025} a^{15} + \frac{13}{60025} a^{14} + \frac{2}{8575} a^{13} - \frac{8}{8575} a^{11} - \frac{11}{8575} a^{10} - \frac{568}{60025} a^{9} + \frac{561}{60025} a^{8} - \frac{103}{2401} a^{7} + \frac{74}{1715} a^{6} + \frac{108}{1715} a^{5} - \frac{89}{1715} a^{4} + \frac{514}{1715} a^{3} - \frac{499}{2401} a^{2} - \frac{313}{2401} a + \frac{765}{2401}$, $\frac{1}{60025} a^{17} - \frac{16}{60025} a^{15} + \frac{13}{60025} a^{14} + \frac{1}{8575} a^{13} + \frac{2}{8575} a^{12} - \frac{2}{1715} a^{11} + \frac{76}{60025} a^{10} - \frac{79}{8575} a^{9} + \frac{443}{60025} a^{8} + \frac{444}{12005} a^{7} - \frac{93}{1715} a^{6} - \frac{22}{1715} a^{5} - \frac{38}{1715} a^{4} - \frac{3111}{12005} a^{3} + \frac{145}{343} a^{2} - \frac{948}{2401} a + \frac{310}{2401}$, $\frac{1}{813853530168047743786475} a^{18} - \frac{9}{813853530168047743786475} a^{17} + \frac{4923756914122721637}{813853530168047743786475} a^{16} - \frac{39390055312981772892}{813853530168047743786475} a^{15} + \frac{219688575173260225049}{813853530168047743786475} a^{14} + \frac{2053189152876443667}{16609255717715260077275} a^{13} - \frac{4394754542758846473}{116264790024006820540925} a^{12} + \frac{406301827215958287508}{813853530168047743786475} a^{11} + \frac{1306747312279172454182}{813853530168047743786475} a^{10} + \frac{7827339797538679345721}{813853530168047743786475} a^{9} - \frac{6723341194845243373199}{813853530168047743786475} a^{8} + \frac{5540065612147771661346}{162770706033609548757295} a^{7} - \frac{258057223512224740683}{4650591600960272821637} a^{6} - \frac{1495148941250970085456}{23252958004801364108185} a^{5} - \frac{2623114534511100092186}{162770706033609548757295} a^{4} - \frac{1575561274033072170902}{162770706033609548757295} a^{3} - \frac{11999610281824770371754}{32554141206721909751459} a^{2} + \frac{12091295706999087083959}{32554141206721909751459} a - \frac{15267237475252763158192}{32554141206721909751459}$, $\frac{1}{28010894212300631875070563611225} a^{19} + \frac{17208796}{28010894212300631875070563611225} a^{18} - \frac{176118061707521770916629106}{28010894212300631875070563611225} a^{17} + \frac{181774106210558158758328079}{28010894212300631875070563611225} a^{16} - \frac{2168180959310088571303165166}{28010894212300631875070563611225} a^{15} - \frac{906994082783993902067614399}{4001556316042947410724366230175} a^{14} - \frac{909185231545504820941261139}{4001556316042947410724366230175} a^{13} + \frac{2005263770621839545255080516}{28010894212300631875070563611225} a^{12} - \frac{9318913693520713610480512349}{28010894212300631875070563611225} a^{11} + \frac{31950610348059560013732037726}{28010894212300631875070563611225} a^{10} + \frac{108732219052354285357125067217}{28010894212300631875070563611225} a^{9} + \frac{49461587852582840183189805187}{28010894212300631875070563611225} a^{8} + \frac{3766790271526216108767646234}{114330180458369926020696178005} a^{7} + \frac{20857225183144534886574664751}{800311263208589482144873246035} a^{6} - \frac{28503528128857246701650559029}{5602178842460126375014112722245} a^{5} + \frac{215889970044588251372105998193}{5602178842460126375014112722245} a^{4} + \frac{1625374833275157649478189767936}{5602178842460126375014112722245} a^{3} + \frac{525273949820806974222222233113}{1120435768492025275002822544449} a^{2} - \frac{235290638304328668790158349484}{1120435768492025275002822544449} a - \frac{26046764601905154274359986110}{160062252641717896428974649207}$

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:  \( 6797274688.460805 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{14}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{14})\), 5.1.50000.1, 10.2.86051840000000000.1, 10.2.430259200000000000.1, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
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}$