Properties

Label 20.0.68204734581...0416.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 7^{10}\cdot 11^{9}$
Root discriminant $11.01$
Ramified primes $2, 7, 11$
Class number $1$
Class group Trivial
Galois Group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 7, 4, 8, 1, -1, -16, 9, -5, 7, 6, -1, -11, 3, -1, 6, -3, 1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 3*x^17 + 6*x^16 - x^15 + 3*x^14 - 11*x^13 - x^12 + 6*x^11 + 7*x^10 - 5*x^9 + 9*x^8 - 16*x^7 - x^6 + x^5 + 8*x^4 + 4*x^3 + 7*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 3*x^17 + 6*x^16 - x^15 + 3*x^14 - 11*x^13 - x^12 + 6*x^11 + 7*x^10 - 5*x^9 + 9*x^8 - 16*x^7 - x^6 + x^5 + 8*x^4 + 4*x^3 + 7*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{19} \) \(\mathstrut +\mathstrut x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut +\mathstrut 3 x^{14} \) \(\mathstrut -\mathstrut 11 x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 7 x^{10} \) \(\mathstrut -\mathstrut 5 x^{9} \) \(\mathstrut +\mathstrut 9 x^{8} \) \(\mathstrut -\mathstrut 16 x^{7} \) \(\mathstrut -\mathstrut x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut 8 x^{4} \) \(\mathstrut +\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(682047345811660860416=2^{10}\cdot 7^{10}\cdot 11^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.01$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7318797294937} a^{19} - \frac{1410639547654}{7318797294937} a^{18} - \frac{390299120196}{7318797294937} a^{17} + \frac{1624629629653}{7318797294937} a^{16} + \frac{769075695028}{7318797294937} a^{15} + \frac{1892297014608}{7318797294937} a^{14} - \frac{2075855182967}{7318797294937} a^{13} + \frac{3218462238322}{7318797294937} a^{12} - \frac{2340064898349}{7318797294937} a^{11} + \frac{387515972805}{7318797294937} a^{10} - \frac{3103520983785}{7318797294937} a^{9} - \frac{2368183457789}{7318797294937} a^{8} + \frac{696462083350}{7318797294937} a^{7} - \frac{475763897942}{7318797294937} a^{6} - \frac{2157400529077}{7318797294937} a^{5} - \frac{2312566444157}{7318797294937} a^{4} - \frac{773286184942}{7318797294937} a^{3} + \frac{2887929273923}{7318797294937} a^{2} + \frac{1542057169862}{7318797294937} a + \frac{1066398384319}{7318797294937}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{2106215076653}{7318797294937} a^{19} - \frac{4584523748496}{7318797294937} a^{18} + \frac{2341536757477}{7318797294937} a^{17} - \frac{5579517260742}{7318797294937} a^{16} + \frac{12609847207926}{7318797294937} a^{15} - \frac{1895050340710}{7318797294937} a^{14} + \frac{2845250513083}{7318797294937} a^{13} - \frac{21659253142772}{7318797294937} a^{12} - \frac{2388328919342}{7318797294937} a^{11} + \frac{19757118415479}{7318797294937} a^{10} + \frac{10032415022443}{7318797294937} a^{9} - \frac{11899710633383}{7318797294937} a^{8} + \frac{18371436508419}{7318797294937} a^{7} - \frac{36044244464268}{7318797294937} a^{6} - \frac{1930390784297}{7318797294937} a^{5} + \frac{11816936507882}{7318797294937} a^{4} + \frac{10828375627315}{7318797294937} a^{3} + \frac{7553756687004}{7318797294937} a^{2} + \frac{13025644109670}{7318797294937} a - \frac{3583074871221}{7318797294937} \),  \( \frac{1575063113903}{7318797294937} a^{19} - \frac{2124250172305}{7318797294937} a^{18} - \frac{841253068699}{7318797294937} a^{17} - \frac{2712472715483}{7318797294937} a^{16} + \frac{5459915654218}{7318797294937} a^{15} + \frac{5905304600938}{7318797294937} a^{14} + \frac{84273292878}{7318797294937} a^{13} - \frac{11262887351996}{7318797294937} a^{12} - \frac{14193966104692}{7318797294937} a^{11} + \frac{14727421444450}{7318797294937} a^{10} + \frac{12259974644961}{7318797294937} a^{9} - \frac{4450847808615}{7318797294937} a^{8} + \frac{7265041045023}{7318797294937} a^{7} - \frac{9463981934319}{7318797294937} a^{6} - \frac{19370229116326}{7318797294937} a^{5} + \frac{4171160877881}{7318797294937} a^{4} + \frac{9322872223434}{7318797294937} a^{3} + \frac{16739336144664}{7318797294937} a^{2} + \frac{8250754264433}{7318797294937} a + \frac{1280811137671}{7318797294937} \),  \( \frac{385853941635}{7318797294937} a^{19} - \frac{422785868063}{7318797294937} a^{18} + \frac{195299734775}{7318797294937} a^{17} - \frac{1539098410995}{7318797294937} a^{16} + \frac{1454302134857}{7318797294937} a^{15} - \frac{159667304620}{7318797294937} a^{14} + \frac{3906466635892}{7318797294937} a^{13} - \frac{3381830646422}{7318797294937} a^{12} - \frac{1161582511379}{7318797294937} a^{11} - \frac{5244575519896}{7318797294937} a^{10} + \frac{3081700759260}{7318797294937} a^{9} + \frac{1831697618834}{7318797294937} a^{8} + \frac{9928988502929}{7318797294937} a^{7} - \frac{3152210625795}{7318797294937} a^{6} - \frac{7928101691864}{7318797294937} a^{5} - \frac{6963984056903}{7318797294937} a^{4} + \frac{1630525561115}{7318797294937} a^{3} + \frac{2725051955247}{7318797294937} a^{2} + \frac{13170573316858}{7318797294937} a + \frac{6108870975866}{7318797294937} \),  \( \frac{1502146542462}{7318797294937} a^{19} - \frac{2114075817261}{7318797294937} a^{18} - \frac{84101538792}{7318797294937} a^{17} - \frac{4370683425570}{7318797294937} a^{16} + \frac{6696150553323}{7318797294937} a^{15} + \frac{4197261632137}{7318797294937} a^{14} + \frac{5029978871414}{7318797294937} a^{13} - \frac{14435966629516}{7318797294937} a^{12} - \frac{13992749861060}{7318797294937} a^{11} + \frac{6705265995858}{7318797294937} a^{10} + \frac{18161887171938}{7318797294937} a^{9} + \frac{6272867379371}{7318797294937} a^{8} + \frac{5730371471389}{7318797294937} a^{7} - \frac{22911232576548}{7318797294937} a^{6} - \frac{12923372677827}{7318797294937} a^{5} + \frac{2925839897055}{7318797294937} a^{4} + \frac{10850872252409}{7318797294937} a^{3} + \frac{18206174842068}{7318797294937} a^{2} + \frac{12170998920654}{7318797294937} a - \frac{1862428255597}{7318797294937} \),  \( \frac{1276071209298}{7318797294937} a^{19} - \frac{2009033949317}{7318797294937} a^{18} + \frac{331055936591}{7318797294937} a^{17} - \frac{3855827112444}{7318797294937} a^{16} + \frac{7153710309456}{7318797294937} a^{15} + \frac{363871939408}{7318797294937} a^{14} + \frac{5994111973458}{7318797294937} a^{13} - \frac{15872433965020}{7318797294937} a^{12} - \frac{3469195770293}{7318797294937} a^{11} + \frac{2402285854808}{7318797294937} a^{10} + \frac{16865300850941}{7318797294937} a^{9} - \frac{5957249791935}{7318797294937} a^{8} + \frac{11052100605773}{7318797294937} a^{7} - \frac{14573436761160}{7318797294937} a^{6} - \frac{6504408337271}{7318797294937} a^{5} - \frac{8130284144190}{7318797294937} a^{4} + \frac{13828114233335}{7318797294937} a^{3} + \frac{4381025078667}{7318797294937} a^{2} + \frac{15814584688647}{7318797294937} a - \frac{2712072861533}{7318797294937} \),  \( \frac{1116357147027}{7318797294937} a^{19} - \frac{1470722247067}{7318797294937} a^{18} - \frac{430601883587}{7318797294937} a^{17} - \frac{2921555522479}{7318797294937} a^{16} + \frac{5207787319646}{7318797294937} a^{15} + \frac{2921772079039}{7318797294937} a^{14} + \frac{4081463105331}{7318797294937} a^{13} - \frac{12651825800880}{7318797294937} a^{12} - \frac{8500949166336}{7318797294937} a^{11} + \frac{3073720095302}{7318797294937} a^{10} + \frac{16976378472586}{7318797294937} a^{9} - \frac{80347822047}{7318797294937} a^{8} + \frac{6215314410966}{7318797294937} a^{7} - \frac{12868472086278}{7318797294937} a^{6} - \frac{14080994735612}{7318797294937} a^{5} - \frac{5138363723663}{7318797294937} a^{4} + \frac{14349070953872}{7318797294937} a^{3} + \frac{12473753847211}{7318797294937} a^{2} + \frac{13522790633328}{7318797294937} a - \frac{1115143000129}{7318797294937} \),  \( \frac{2299679184624}{7318797294937} a^{19} - \frac{3529666032577}{7318797294937} a^{18} + \frac{395855169313}{7318797294937} a^{17} - \frac{6659013286166}{7318797294937} a^{16} + \frac{11391616201287}{7318797294937} a^{15} + \frac{3450688600957}{7318797294937} a^{14} + \frac{7935553442551}{7318797294937} a^{13} - \frac{24397373819874}{7318797294937} a^{12} - \frac{13711536237294}{7318797294937} a^{11} + \frac{10208658661310}{7318797294937} a^{10} + \frac{26945503313214}{7318797294937} a^{9} - \frac{2544142523632}{7318797294937} a^{8} + \frac{11672205681572}{7318797294937} a^{7} - \frac{32735874403983}{7318797294937} a^{6} - \frac{12113141404582}{7318797294937} a^{5} + \frac{923295038153}{7318797294937} a^{4} + \frac{21167869019196}{7318797294937} a^{3} + \frac{16059665364905}{7318797294937} a^{2} + \frac{21922765551135}{7318797294937} a - \frac{5397460544715}{7318797294937} \),  \( \frac{636567273296}{7318797294937} a^{19} - \frac{1178351227650}{7318797294937} a^{18} - \frac{283502569980}{7318797294937} a^{17} - \frac{766982121942}{7318797294937} a^{16} + \frac{4160955301312}{7318797294937} a^{15} + \frac{1314589751840}{7318797294937} a^{14} - \frac{1938566683843}{7318797294937} a^{13} - \frac{9799329433089}{7318797294937} a^{12} - \frac{1854180251097}{7318797294937} a^{11} + \frac{12994615475292}{7318797294937} a^{10} + \frac{12364037481681}{7318797294937} a^{9} - \frac{10837648749084}{7318797294937} a^{8} - \frac{8639488848372}{7318797294937} a^{7} - \frac{7419981061377}{7318797294937} a^{6} + \frac{1560573715595}{7318797294937} a^{5} + \frac{10258045414837}{7318797294937} a^{4} + \frac{11340214549058}{7318797294937} a^{3} + \frac{1478915311480}{7318797294937} a^{2} - \frac{7883441203917}{7318797294937} a - \frac{3676123184321}{7318797294937} \),  \( \frac{1318135993969}{7318797294937} a^{19} - \frac{2012933247783}{7318797294937} a^{18} + \frac{435102521240}{7318797294937} a^{17} - \frac{4157965303004}{7318797294937} a^{16} + \frac{6212098972939}{7318797294937} a^{15} + \frac{2233206184812}{7318797294937} a^{14} + \frac{5013851656476}{7318797294937} a^{13} - \frac{12089360487025}{7318797294937} a^{12} - \frac{9865619736591}{7318797294937} a^{11} + \frac{5430805075041}{7318797294937} a^{10} + \frac{11367883523744}{7318797294937} a^{9} + \frac{4520560918571}{7318797294937} a^{8} + \frac{6296678413903}{7318797294937} a^{7} - \frac{18952840846028}{7318797294937} a^{6} - \frac{5833106520361}{7318797294937} a^{5} + \frac{236776028583}{7318797294937} a^{4} + \frac{6130347134914}{7318797294937} a^{3} + \frac{13596043089389}{7318797294937} a^{2} + \frac{13569212362923}{7318797294937} a + \frac{1445953083920}{7318797294937} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 87.3510823497 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.2156.1, 10.0.246071287.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$