LMFDB - Number field 21.3.270061427296836406180775407.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1)

gp: K = bnfinit(y^21 - 4*y^18 - y^17 + y^16 + 7*y^15 + y^14 - 6*y^13 - 11*y^12 + 2*y^11 + 10*y^10 + 14*y^9 - 2*y^8 - 5*y^7 - 6*y^6 + 2*y^5 - 2*y^4 - 3*y^3 + 2*y^2 - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1)

$ x^{21} - 4 x^{18} - x^{17} + x^{16} + 7 x^{15} + x^{14} - 6 x^{13} - 11 x^{12} + 2 x^{11} + 10 x^{10} + \cdots - 1 $ x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$21$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[3, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-270061427296836406180775407$ -270061427296836406180775407 $\medspace = -\,193327^{5}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$18.14$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$193327^{1/2}\approx 439.68966328536766$
Ramified primes:	$193327$ 193327	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-193327}) $
$\card{ \Aut(K/\Q) }$:	$1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{1}{3}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{2848767}a^{20}+\frac{135774}{949589}a^{19}-\frac{28807}{2848767}a^{18}+\frac{346415}{2848767}a^{17}-\frac{259079}{949589}a^{16}-\frac{353425}{2848767}a^{15}-\frac{1035032}{2848767}a^{14}-\frac{376795}{2848767}a^{13}+\frac{1378718}{2848767}a^{12}+\frac{936119}{2848767}a^{11}-\frac{283895}{949589}a^{10}+\frac{463043}{2848767}a^{9}-\frac{715909}{2848767}a^{8}-\frac{947635}{2848767}a^{7}+\frac{151601}{2848767}a^{6}-\frac{400565}{2848767}a^{5}+\frac{131547}{949589}a^{4}-\frac{154840}{949589}a^{3}+\frac{63721}{949589}a^{2}-\frac{452923}{2848767}a+\frac{216238}{949589}$ 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, 1/3*a^19 - 1/3*a^17 - 1/3*a^16 - 1/3*a^14 + 1/3*a^13 - 1/3*a^12 - 1/3*a^11 - 1/3*a^10 - 1/3*a^8 - 1/3*a^7 - 1/3*a^6 - 1/3*a^5 + 1/3*a^4 - 1/3, 1/2848767*a^20 + 135774/949589*a^19 - 28807/2848767*a^18 + 346415/2848767*a^17 - 259079/949589*a^16 - 353425/2848767*a^15 - 1035032/2848767*a^14 - 376795/2848767*a^13 + 1378718/2848767*a^12 + 936119/2848767*a^11 - 283895/949589*a^10 + 463043/2848767*a^9 - 715909/2848767*a^8 - 947635/2848767*a^7 + 151601/2848767*a^6 - 400565/2848767*a^5 + 131547/949589*a^4 - 154840/949589*a^3 + 63721/949589*a^2 - 452923/2848767*a + 216238/949589

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$a^{20}-4a^{17}-a^{16}+a^{15}+7a^{14}+a^{13}-6a^{12}-11a^{11}+2a^{10}+10a^{9}+14a^{8}-2a^{7}-5a^{6}-6a^{5}+2a^{4}-2a^{3}-3a^{2}+2a$, $\frac{398921}{949589}a^{20}-\frac{765697}{2848767}a^{19}+\frac{208831}{949589}a^{18}-\frac{5356454}{2848767}a^{17}+\frac{1421119}{2848767}a^{16}-\frac{326828}{949589}a^{15}+\frac{9067168}{2848767}a^{14}-\frac{2036566}{2848767}a^{13}-\frac{2408621}{2848767}a^{12}-\frac{11569814}{2848767}a^{11}+\frac{5694754}{2848767}a^{10}+\frac{1625145}{949589}a^{9}+\frac{17961997}{2848767}a^{8}-\frac{4161161}{2848767}a^{7}+\frac{2942812}{2848767}a^{6}-\frac{11751293}{2848767}a^{5}+\frac{1313276}{2848767}a^{4}-\frac{3035871}{949589}a^{3}+\frac{291300}{949589}a^{2}+\frac{651714}{949589}a-\frac{1414715}{2848767}$, $\frac{493672}{2848767}a^{20}-\frac{1500256}{2848767}a^{19}-\frac{164440}{2848767}a^{18}-\frac{914751}{949589}a^{17}+\frac{5030911}{2848767}a^{16}+\frac{2405849}{2848767}a^{15}+\frac{4674629}{2848767}a^{14}-\frac{7647920}{2848767}a^{13}-\frac{1761267}{949589}a^{12}-\frac{246554}{949589}a^{11}+\frac{15475990}{2848767}a^{10}+\frac{6299816}{2848767}a^{9}+\frac{467628}{949589}a^{8}-\frac{6696235}{949589}a^{7}-\frac{1812036}{949589}a^{6}-\frac{7210498}{2848767}a^{5}+\frac{5532512}{2848767}a^{4}-\frac{1106747}{949589}a^{3}+\frac{1188298}{949589}a^{2}+\frac{4318574}{2848767}a-\frac{1700387}{2848767}$, $\frac{3163}{949589}a^{20}+\frac{1200817}{2848767}a^{19}+\frac{44003}{949589}a^{18}+\frac{604406}{2848767}a^{17}-\frac{4492075}{2848767}a^{16}-\frac{217022}{949589}a^{15}-\frac{769210}{2848767}a^{14}+\frac{7391542}{2848767}a^{13}-\frac{782140}{2848767}a^{12}-\frac{4370260}{2848767}a^{11}-\frac{12981232}{2848767}a^{10}+\frac{338771}{949589}a^{9}+\frac{4847150}{2848767}a^{8}+\frac{13793561}{2848767}a^{7}-\frac{1984624}{2848767}a^{6}+\frac{243482}{2848767}a^{5}-\frac{5178512}{2848767}a^{4}+\frac{1636601}{949589}a^{3}-\frac{239624}{949589}a^{2}+\frac{334352}{949589}a+\frac{3260015}{2848767}$, $\frac{551705}{2848767}a^{20}-\frac{184006}{949589}a^{19}+\frac{305158}{2848767}a^{18}-\frac{2041688}{2848767}a^{17}+\frac{754941}{949589}a^{16}+\frac{366457}{2848767}a^{15}+\frac{3015590}{2848767}a^{14}-\frac{5157485}{2848767}a^{13}-\frac{3662480}{2848767}a^{12}-\frac{2831603}{2848767}a^{11}+\frac{2717041}{949589}a^{10}+\frac{2806357}{2848767}a^{9}-\frac{425363}{2848767}a^{8}-\frac{12096602}{2848767}a^{7}-\frac{769415}{2848767}a^{6}+\frac{2235467}{2848767}a^{5}+\frac{3747899}{949589}a^{4}-\frac{26171}{949589}a^{3}+\frac{459936}{949589}a^{2}+\frac{2562457}{2848767}a-\frac{1078636}{949589}$, $\frac{1027450}{2848767}a^{20}+\frac{124820}{2848767}a^{19}+\frac{936980}{2848767}a^{18}-\frac{1551862}{949589}a^{17}-\frac{1042265}{2848767}a^{16}-\frac{2733061}{2848767}a^{15}+\frac{8689412}{2848767}a^{14}+\frac{1815838}{2848767}a^{13}-\frac{258358}{949589}a^{12}-\frac{4895083}{949589}a^{11}-\frac{3245915}{2848767}a^{10}+\frac{3743816}{2848767}a^{9}+\frac{7108066}{949589}a^{8}+\frac{1018307}{949589}a^{7}+\frac{771123}{949589}a^{6}-\frac{14334184}{2848767}a^{5}+\frac{1291628}{2848767}a^{4}-\frac{964885}{949589}a^{3}+\frac{727845}{949589}a^{2}-\frac{1100599}{2848767}a-\frac{67745}{2848767}$, $\frac{1215101}{949589}a^{20}-\frac{3235394}{2848767}a^{19}+\frac{335211}{949589}a^{18}-\frac{15696193}{2848767}a^{17}+\frac{9138065}{2848767}a^{16}+\frac{755969}{949589}a^{15}+\frac{25206674}{2848767}a^{14}-\frac{16977026}{2848767}a^{13}-\frac{17300665}{2848767}a^{12}-\frac{25572712}{2848767}a^{11}+\frac{37047479}{2848767}a^{10}+\frac{7781898}{949589}a^{9}+\frac{31231493}{2848767}a^{8}-\frac{43626121}{2848767}a^{7}-\frac{2629405}{2848767}a^{6}-\frac{18681853}{2848767}a^{5}+\frac{23258275}{2848767}a^{4}-\frac{6813276}{949589}a^{3}-\frac{411183}{949589}a^{2}+\frac{3257740}{949589}a-\frac{7144303}{2848767}$, $\frac{1379024}{2848767}a^{20}+\frac{230914}{2848767}a^{19}+\frac{511447}{2848767}a^{18}-\frac{1861583}{949589}a^{17}-\frac{1733485}{2848767}a^{16}-\frac{255005}{2848767}a^{15}+\frac{10349734}{2848767}a^{14}+\frac{1954364}{2848767}a^{13}-\frac{2191294}{949589}a^{12}-\frac{6195014}{949589}a^{11}-\frac{598324}{2848767}a^{10}+\frac{10530817}{2848767}a^{9}+\frac{7498198}{949589}a^{8}-\frac{423425}{949589}a^{7}-\frac{1931450}{949589}a^{6}-\frac{11877671}{2848767}a^{5}+\frac{4256950}{2848767}a^{4}-\frac{644853}{949589}a^{3}+\frac{671011}{949589}a^{2}+\frac{3326365}{2848767}a+\frac{3370016}{2848767}$, $\frac{69609}{949589}a^{20}-\frac{500853}{949589}a^{19}+\frac{305505}{949589}a^{18}-\frac{261331}{949589}a^{17}+\frac{2042120}{949589}a^{16}-\frac{558602}{949589}a^{15}-\frac{325880}{949589}a^{14}-\frac{3523742}{949589}a^{13}+\frac{968977}{949589}a^{12}+\frac{2459880}{949589}a^{11}+\frac{4547228}{949589}a^{10}-\frac{3737596}{949589}a^{9}-\frac{4026806}{949589}a^{8}-\frac{5472775}{949589}a^{7}+\frac{4759397}{949589}a^{6}+\frac{2701489}{949589}a^{5}+\frac{2753955}{949589}a^{4}-\frac{3166408}{949589}a^{3}+\frac{1024199}{949589}a^{2}+\frac{736871}{949589}a-\frac{422480}{949589}$, $\frac{161320}{949589}a^{20}+\frac{475432}{2848767}a^{19}+\frac{143326}{949589}a^{18}-\frac{833728}{2848767}a^{17}-\frac{2323018}{2848767}a^{16}-\frac{247851}{949589}a^{15}-\frac{191686}{2848767}a^{14}+\frac{3464182}{2848767}a^{13}-\frac{1440172}{2848767}a^{12}-\frac{2861482}{2848767}a^{11}-\frac{6669616}{2848767}a^{10}-\frac{372336}{949589}a^{9}+\frac{2370023}{2848767}a^{8}+\frac{9363086}{2848767}a^{7}+\frac{2624231}{2848767}a^{6}+\frac{4955306}{2848767}a^{5}-\frac{3225977}{2848767}a^{4}+\frac{449535}{949589}a^{3}-\frac{437204}{949589}a^{2}+\frac{1536834}{949589}a-\frac{1487740}{2848767}$, $\frac{1500256}{2848767}a^{20}+\frac{164440}{2848767}a^{19}+\frac{769565}{2848767}a^{18}-\frac{5524583}{2848767}a^{17}-\frac{1912177}{2848767}a^{16}-\frac{1218925}{2848767}a^{15}+\frac{2713864}{949589}a^{14}+\frac{773923}{949589}a^{13}-\frac{4690730}{2848767}a^{12}-\frac{14488646}{2848767}a^{11}-\frac{1363096}{2848767}a^{10}+\frac{5508524}{2848767}a^{9}+\frac{19101361}{2848767}a^{8}+\frac{2967748}{2848767}a^{7}+\frac{4248466}{2848767}a^{6}-\frac{1515056}{949589}a^{5}+\frac{2332897}{2848767}a^{4}-\frac{1681970}{949589}a^{3}-\frac{2059999}{949589}a^{2}+\frac{1700387}{2848767}a-\frac{3342439}{2848767}$ a^20 - 4a^17 - a^16 + a^15 + 7a^14 + a^13 - 6a^12 - 11a^11 + 2a^10 + 10a^9 + 14a^8 - 2a^7 - 5a^6 - 6a^5 + 2a^4 - 2a^3 - 3a^2 + 2a, 398921/949589a^20 - 765697/2848767a^19 + 208831/949589a^18 - 5356454/2848767a^17 + 1421119/2848767a^16 - 326828/949589a^15 + 9067168/2848767a^14 - 2036566/2848767a^13 - 2408621/2848767a^12 - 11569814/2848767a^11 + 5694754/2848767a^10 + 1625145/949589a^9 + 17961997/2848767a^8 - 4161161/2848767a^7 + 2942812/2848767a^6 - 11751293/2848767a^5 + 1313276/2848767a^4 - 3035871/949589a^3 + 291300/949589a^2 + 651714/949589a - 1414715/2848767, 493672/2848767a^20 - 1500256/2848767a^19 - 164440/2848767a^18 - 914751/949589a^17 + 5030911/2848767a^16 + 2405849/2848767a^15 + 4674629/2848767a^14 - 7647920/2848767a^13 - 1761267/949589a^12 - 246554/949589a^11 + 15475990/2848767a^10 + 6299816/2848767a^9 + 467628/949589a^8 - 6696235/949589a^7 - 1812036/949589a^6 - 7210498/2848767a^5 + 5532512/2848767a^4 - 1106747/949589a^3 + 1188298/949589a^2 + 4318574/2848767a - 1700387/2848767, 3163/949589a^20 + 1200817/2848767a^19 + 44003/949589a^18 + 604406/2848767a^17 - 4492075/2848767a^16 - 217022/949589a^15 - 769210/2848767a^14 + 7391542/2848767a^13 - 782140/2848767a^12 - 4370260/2848767a^11 - 12981232/2848767a^10 + 338771/949589a^9 + 4847150/2848767a^8 + 13793561/2848767a^7 - 1984624/2848767a^6 + 243482/2848767a^5 - 5178512/2848767a^4 + 1636601/949589a^3 - 239624/949589a^2 + 334352/949589a + 3260015/2848767, 551705/2848767a^20 - 184006/949589a^19 + 305158/2848767a^18 - 2041688/2848767a^17 + 754941/949589a^16 + 366457/2848767a^15 + 3015590/2848767a^14 - 5157485/2848767a^13 - 3662480/2848767a^12 - 2831603/2848767a^11 + 2717041/949589a^10 + 2806357/2848767a^9 - 425363/2848767a^8 - 12096602/2848767a^7 - 769415/2848767a^6 + 2235467/2848767a^5 + 3747899/949589a^4 - 26171/949589a^3 + 459936/949589a^2 + 2562457/2848767a - 1078636/949589, 1027450/2848767a^20 + 124820/2848767a^19 + 936980/2848767a^18 - 1551862/949589a^17 - 1042265/2848767a^16 - 2733061/2848767a^15 + 8689412/2848767a^14 + 1815838/2848767a^13 - 258358/949589a^12 - 4895083/949589a^11 - 3245915/2848767a^10 + 3743816/2848767a^9 + 7108066/949589a^8 + 1018307/949589a^7 + 771123/949589a^6 - 14334184/2848767a^5 + 1291628/2848767a^4 - 964885/949589a^3 + 727845/949589a^2 - 1100599/2848767a - 67745/2848767, 1215101/949589a^20 - 3235394/2848767a^19 + 335211/949589a^18 - 15696193/2848767a^17 + 9138065/2848767a^16 + 755969/949589a^15 + 25206674/2848767a^14 - 16977026/2848767a^13 - 17300665/2848767a^12 - 25572712/2848767a^11 + 37047479/2848767a^10 + 7781898/949589a^9 + 31231493/2848767a^8 - 43626121/2848767a^7 - 2629405/2848767a^6 - 18681853/2848767a^5 + 23258275/2848767a^4 - 6813276/949589a^3 - 411183/949589a^2 + 3257740/949589a - 7144303/2848767, 1379024/2848767a^20 + 230914/2848767a^19 + 511447/2848767a^18 - 1861583/949589a^17 - 1733485/2848767a^16 - 255005/2848767a^15 + 10349734/2848767a^14 + 1954364/2848767a^13 - 2191294/949589a^12 - 6195014/949589a^11 - 598324/2848767a^10 + 10530817/2848767a^9 + 7498198/949589a^8 - 423425/949589a^7 - 1931450/949589a^6 - 11877671/2848767a^5 + 4256950/2848767a^4 - 644853/949589a^3 + 671011/949589a^2 + 3326365/2848767a + 3370016/2848767, 69609/949589a^20 - 500853/949589a^19 + 305505/949589a^18 - 261331/949589a^17 + 2042120/949589a^16 - 558602/949589a^15 - 325880/949589a^14 - 3523742/949589a^13 + 968977/949589a^12 + 2459880/949589a^11 + 4547228/949589a^10 - 3737596/949589a^9 - 4026806/949589a^8 - 5472775/949589a^7 + 4759397/949589a^6 + 2701489/949589a^5 + 2753955/949589a^4 - 3166408/949589a^3 + 1024199/949589a^2 + 736871/949589a - 422480/949589, 161320/949589a^20 + 475432/2848767a^19 + 143326/949589a^18 - 833728/2848767a^17 - 2323018/2848767a^16 - 247851/949589a^15 - 191686/2848767a^14 + 3464182/2848767a^13 - 1440172/2848767a^12 - 2861482/2848767a^11 - 6669616/2848767a^10 - 372336/949589a^9 + 2370023/2848767a^8 + 9363086/2848767a^7 + 2624231/2848767a^6 + 4955306/2848767a^5 - 3225977/2848767a^4 + 449535/949589a^3 - 437204/949589a^2 + 1536834/949589a - 1487740/2848767, 1500256/2848767a^20 + 164440/2848767a^19 + 769565/2848767a^18 - 5524583/2848767a^17 - 1912177/2848767a^16 - 1218925/2848767a^15 + 2713864/949589a^14 + 773923/949589a^13 - 4690730/2848767a^12 - 14488646/2848767a^11 - 1363096/2848767a^10 + 5508524/2848767a^9 + 19101361/2848767a^8 + 2967748/2848767a^7 + 4248466/2848767a^6 - 1515056/949589a^5 + 2332897/2848767a^4 - 1681970/949589a^3 - 2059999/949589a^2 + 1700387/2848767a - 3342439/2848767 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 75073.7253 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 75073.7253 \cdot 1}{2\cdot\sqrt{270061427296836406180775407}}\cr\approx \mathstrut & 0.278891639 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^18 - x^17 + x^16 + 7*x^15 + x^14 - 6*x^13 - 11*x^12 + 2*x^11 + 10*x^10 + 14*x^9 - 2*x^8 - 5*x^7 - 6*x^6 + 2*x^5 - 2*x^4 - 3*x^3 + 2*x^2 - 1);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$S_7$ (as 21T38):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A non-solvable group of order 5040

The 15 conjugacy class representatives for $S_7$

Character table for $S_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 7 sibling:	7.1.193327.1
Degree 14 sibling:	deg 14
Degree 30 sibling:	data not computed
Degree 35 sibling:	data not computed
Degree 42 siblings:	data not computed
Minimal sibling:	7.1.193327.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.7.0.1}{7} }^{3}$	${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$	${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.7.0.1}{7} }^{3}$	${\href{/padicField/13.3.0.1}{3} }^{7}$	${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$	${\href{/padicField/19.7.0.1}{7} }^{3}$	${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.7.0.1}{7} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$193327$ 193327	$\Q_{193327}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $8$	$2$	$4$	$4$

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