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Properties

Label 20.0.27511800617...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 19^{10}$
Root discriminant $66.37$
Ramified primes $5, 11, 19$
Class number $45100$ (GRH)
Class group $[5, 9020]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![305078741, -272454303, 390090877, -256367111, 207253409, -106576405, 62549497, -26337315, 12400559, -4438083, 1770300, -551591, 190246, -51533, 15425, -3670, 990, -216, 51, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 51*x^18 - 216*x^17 + 990*x^16 - 3670*x^15 + 15425*x^14 - 51533*x^13 + 190246*x^12 - 551591*x^11 + 1770300*x^10 - 4438083*x^9 + 12400559*x^8 - 26337315*x^7 + 62549497*x^6 - 106576405*x^5 + 207253409*x^4 - 256367111*x^3 + 390090877*x^2 - 272454303*x + 305078741)
 
gp: K = bnfinit(x^20 - 8*x^19 + 51*x^18 - 216*x^17 + 990*x^16 - 3670*x^15 + 15425*x^14 - 51533*x^13 + 190246*x^12 - 551591*x^11 + 1770300*x^10 - 4438083*x^9 + 12400559*x^8 - 26337315*x^7 + 62549497*x^6 - 106576405*x^5 + 207253409*x^4 - 256367111*x^3 + 390090877*x^2 - 272454303*x + 305078741, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 51 x^{18} - 216 x^{17} + 990 x^{16} - 3670 x^{15} + 15425 x^{14} - 51533 x^{13} + 190246 x^{12} - 551591 x^{11} + 1770300 x^{10} - 4438083 x^{9} + 12400559 x^{8} - 26337315 x^{7} + 62549497 x^{6} - 106576405 x^{5} + 207253409 x^{4} - 256367111 x^{3} + 390090877 x^{2} - 272454303 x + 305078741 \)

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:  \(2751180061734552963011647245712890625=5^{10}\cdot 11^{16}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.37$
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:  $5, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1045=5\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1045}(1,·)$, $\chi_{1045}(324,·)$, $\chi_{1045}(911,·)$, $\chi_{1045}(664,·)$, $\chi_{1045}(531,·)$, $\chi_{1045}(856,·)$, $\chi_{1045}(474,·)$, $\chi_{1045}(476,·)$, $\chi_{1045}(609,·)$, $\chi_{1045}(419,·)$, $\chi_{1045}(229,·)$, $\chi_{1045}(284,·)$, $\chi_{1045}(1006,·)$, $\chi_{1045}(949,·)$, $\chi_{1045}(246,·)$, $\chi_{1045}(951,·)$, $\chi_{1045}(56,·)$, $\chi_{1045}(379,·)$, $\chi_{1045}(894,·)$, $\chi_{1045}(191,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} + \frac{122}{331} a^{17} - \frac{152}{331} a^{16} + \frac{49}{331} a^{15} - \frac{133}{331} a^{14} - \frac{76}{331} a^{13} + \frac{23}{331} a^{12} - \frac{157}{331} a^{11} - \frac{20}{331} a^{10} - \frac{95}{331} a^{9} - \frac{132}{331} a^{8} + \frac{92}{331} a^{7} - \frac{51}{331} a^{6} + \frac{133}{331} a^{5} + \frac{162}{331} a^{4} + \frac{70}{331} a^{3} + \frac{24}{331} a^{2} + \frac{140}{331} a - \frac{138}{331}$, $\frac{1}{60558333527414422040831672375739546536135035717226921491094476589} a^{19} + \frac{89758611858847686891130195819603720814383757670590451502279048}{60558333527414422040831672375739546536135035717226921491094476589} a^{18} - \frac{14977711735227043935071303554491680140701701816175649139239221989}{60558333527414422040831672375739546536135035717226921491094476589} a^{17} + \frac{1039825407608821401873271460750923151167879069747634526497482349}{60558333527414422040831672375739546536135035717226921491094476589} a^{16} + \frac{23385877042182212960478279525712827316565773187793298162926751181}{60558333527414422040831672375739546536135035717226921491094476589} a^{15} + \frac{4654362270548963843040021819023066212407735298145263297003673684}{60558333527414422040831672375739546536135035717226921491094476589} a^{14} - \frac{191214989943403960620790327677382627913443274309325896656879703}{462277355171102458326959331112515622413244547459747492298431119} a^{13} - \frac{27686068756982621505118126970065877474484700273399957084329114724}{60558333527414422040831672375739546536135035717226921491094476589} a^{12} + \frac{16116623662321936625136230371576414344360932941731777503064259998}{60558333527414422040831672375739546536135035717226921491094476589} a^{11} + \frac{22714091965172163298722911082426946365547927112598850520256572971}{60558333527414422040831672375739546536135035717226921491094476589} a^{10} + \frac{29457945405020410794473891893595688636241512561230839283728907955}{60558333527414422040831672375739546536135035717226921491094476589} a^{9} - \frac{69152856336430844945478748137938238039072457950781047553675053}{182955690415149311301606260953895910985302222710655351936841319} a^{8} + \frac{29591501999004210966128923916497177939700543780133637033743743156}{60558333527414422040831672375739546536135035717226921491094476589} a^{7} - \frac{17084363668602977528510418876502227735961878079990750773486116273}{60558333527414422040831672375739546536135035717226921491094476589} a^{6} + \frac{5225064048018982266927912669773942883373502437432824915730546064}{60558333527414422040831672375739546536135035717226921491094476589} a^{5} + \frac{27947635300269218603183405296406715035866999398329830918735859371}{60558333527414422040831672375739546536135035717226921491094476589} a^{4} - \frac{11521619924816251355473854855646515475513042632596403781075815647}{60558333527414422040831672375739546536135035717226921491094476589} a^{3} - \frac{5303800085351994095013915736294849240824674385079007465541968124}{60558333527414422040831672375739546536135035717226921491094476589} a^{2} - \frac{15573910029515770721084176924783792607025831525870276986201107387}{60558333527414422040831672375739546536135035717226921491094476589} a - \frac{111488699609577636716092479318715896299773112722426662512527019}{304313233806102623320762172742409781588618269935813675834645611}$

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

Class group and class number

$C_{5}\times C_{9020}$, which has order $45100$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.1658668159016309375.3, 10.0.530773810885219.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$19$19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$