LMFDB - Number field 18.10.385398969258228918652928.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*x^2 + 1)

gp: K = bnfinit(y^18 - y^17 - 13*y^16 + 4*y^15 + 70*y^14 + 21*y^13 - 184*y^12 - 139*y^11 + 233*y^10 + 255*y^9 - 148*y^8 - 188*y^7 + 84*y^6 + 61*y^5 - 48*y^4 - 5*y^3 + 4*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*x^2 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*x^2 + 1)

$ x^{18} - x^{17} - 13 x^{16} + 4 x^{15} + 70 x^{14} + 21 x^{13} - 184 x^{12} - 139 x^{11} + 233 x^{10} + \cdots + 1 $ x^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*x^2 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[10, 4]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$385398969258228918652928$ 385398969258228918652928 $\medspace = 2^{12}\cdot 37^{6}\cdot 137\cdot 16361^{2}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.43$	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:	$2^{2/3}37^{1/2}137^{1/2}16361^{1/2}\approx 14456.15285607937$
Ramified primes:	$2$, $37$, $137$, $16361$ 2, 37, 137, 16361	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{137}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{9580468}a^{17}+\frac{1171299}{9580468}a^{16}+\frac{340151}{9580468}a^{15}-\frac{1661295}{9580468}a^{14}-\frac{87163}{2395117}a^{13}+\frac{941389}{9580468}a^{12}+\frac{1736875}{9580468}a^{11}+\frac{641573}{4790234}a^{10}-\frac{1886643}{4790234}a^{9}+\frac{1105216}{2395117}a^{8}+\frac{318316}{2395117}a^{7}+\frac{4225505}{9580468}a^{6}+\frac{4359859}{9580468}a^{5}-\frac{2752683}{9580468}a^{4}-\frac{1053497}{4790234}a^{3}-\frac{175189}{2395117}a^{2}-\frac{1638799}{4790234}a+\frac{3463039}{9580468}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/2*a^12 - 1/2*a^11 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^13 - 1/2*a^11 - 1/2*a^10 - 1/2*a^7 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^14 - 1/2*a^9 - 1/2*a^7 - 1/2*a^5 - 1/2*a - 1/2, 1/4*a^15 - 1/4*a^12 + 1/4*a^11 + 1/4*a^10 - 1/4*a^9 - 1/4*a^7 - 1/4*a^6 - 1/2*a^5 - 1/2*a^4 + 1/4*a^3 - 1/2*a^2 + 1/4*a + 1/4, 1/4*a^16 - 1/4*a^13 - 1/4*a^12 - 1/4*a^11 - 1/4*a^10 - 1/2*a^9 + 1/4*a^8 + 1/4*a^7 - 1/2*a^6 - 1/2*a^5 + 1/4*a^4 - 1/4*a^2 + 1/4*a - 1/2, 1/9580468*a^17 + 1171299/9580468*a^16 + 340151/9580468*a^15 - 1661295/9580468*a^14 - 87163/2395117*a^13 + 941389/9580468*a^12 + 1736875/9580468*a^11 + 641573/4790234*a^10 - 1886643/4790234*a^9 + 1105216/2395117*a^8 + 318316/2395117*a^7 + 4225505/9580468*a^6 + 4359859/9580468*a^5 - 2752683/9580468*a^4 - 1053497/4790234*a^3 - 175189/2395117*a^2 - 1638799/4790234*a + 3463039/9580468

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 $1$

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:	$13$	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:	$a$, $\frac{1226575}{9580468}a^{17}-\frac{910355}{9580468}a^{16}-\frac{13458809}{9580468}a^{15}-\frac{4434301}{9580468}a^{14}+\frac{28898129}{4790234}a^{13}+\frac{79740953}{9580468}a^{12}-\frac{77859779}{9580468}a^{11}-\frac{131575363}{4790234}a^{10}-\frac{27436295}{2395117}a^{9}+\frac{68341827}{2395117}a^{8}+\frac{77488806}{2395117}a^{7}-\frac{43378179}{9580468}a^{6}-\frac{171087015}{9580468}a^{5}+\frac{19493941}{9580468}a^{4}+\frac{13813208}{2395117}a^{3}-\frac{21027625}{4790234}a^{2}-\frac{3150941}{4790234}a-\frac{5455267}{9580468}$, $\frac{370110}{2395117}a^{17}-\frac{1432635}{4790234}a^{16}-\frac{16968629}{9580468}a^{15}+\frac{5358439}{2395117}a^{14}+\frac{44970767}{4790234}a^{13}-\frac{46256023}{9580468}a^{12}-\frac{261971145}{9580468}a^{11}-\frac{11296619}{9580468}a^{10}+\frac{417418267}{9580468}a^{9}+\frac{36884181}{2395117}a^{8}-\frac{356526911}{9580468}a^{7}-\frac{158753111}{9580468}a^{6}+\frac{76578297}{4790234}a^{5}+\frac{10622926}{2395117}a^{4}+\frac{32473}{9580468}a^{3}+\frac{8857955}{4790234}a^{2}-\frac{26838171}{9580468}a+\frac{8850501}{9580468}$, $\frac{3309081}{9580468}a^{17}-\frac{582659}{4790234}a^{16}-\frac{45925259}{9580468}a^{15}-\frac{10537751}{9580468}a^{14}+\frac{242476437}{9580468}a^{13}+\frac{46148803}{2395117}a^{12}-\frac{282230303}{4790234}a^{11}-\frac{721557587}{9580468}a^{10}+\frac{249076477}{4790234}a^{9}+\frac{1057105067}{9580468}a^{8}-\frac{57916219}{9580468}a^{7}-\frac{585856155}{9580468}a^{6}+\frac{19574229}{9580468}a^{5}+\frac{27964485}{2395117}a^{4}-\frac{17834405}{2395117}a^{3}+\frac{29246005}{9580468}a^{2}-\frac{26561221}{9580468}a-\frac{8631681}{9580468}$, $\frac{1957829}{4790234}a^{17}-\frac{6412013}{4790234}a^{16}-\frac{31135395}{9580468}a^{15}+\frac{51728147}{4790234}a^{14}+\frac{34810767}{2395117}a^{13}-\frac{337259045}{9580468}a^{12}-\frac{445973139}{9580468}a^{11}+\frac{571856047}{9580468}a^{10}+\frac{869069969}{9580468}a^{9}-\frac{136937512}{2395117}a^{8}-\frac{907348363}{9580468}a^{7}+\frac{367806235}{9580468}a^{6}+\frac{116915537}{2395117}a^{5}-\frac{59231418}{2395117}a^{4}-\frac{69449905}{9580468}a^{3}+\frac{17686567}{4790234}a^{2}+\frac{10966795}{9580468}a+\frac{4108429}{9580468}$, $\frac{4108429}{9580468}a^{17}-\frac{8024087}{9580468}a^{16}-\frac{40585551}{9580468}a^{15}+\frac{47569111}{9580468}a^{14}+\frac{46033434}{2395117}a^{13}-\frac{52966059}{9580468}a^{12}-\frac{418691891}{9580468}a^{11}-\frac{31274623}{2395117}a^{10}+\frac{192703955}{4790234}a^{9}+\frac{89289713}{4790234}a^{8}-\frac{15074361}{2395117}a^{7}+\frac{134963711}{9580468}a^{6}-\frac{22698199}{9580468}a^{5}-\frac{217047979}{9580468}a^{4}+\frac{9930270}{2395117}a^{3}+\frac{12226940}{2395117}a^{2}-\frac{9469709}{4790234}a-\frac{1386327}{9580468}$, $\frac{3443937}{9580468}a^{17}-\frac{9618875}{9580468}a^{16}-\frac{7259242}{2395117}a^{15}+\frac{69751653}{9580468}a^{14}+\frac{32059634}{2395117}a^{13}-\frac{91539645}{4790234}a^{12}-\frac{88436802}{2395117}a^{11}+\frac{210062003}{9580468}a^{10}+\frac{531219777}{9580468}a^{9}-\frac{63297601}{4790234}a^{8}-\frac{382299877}{9580468}a^{7}+\frac{29687367}{2395117}a^{6}+\frac{115203053}{9580468}a^{5}-\frac{121893461}{9580468}a^{4}+\frac{18809023}{9580468}a^{3}-\frac{3047767}{4790234}a^{2}+\frac{12476411}{9580468}a+\frac{323963}{4790234}$, $\frac{1190897}{4790234}a^{17}-\frac{1454889}{2395117}a^{16}-\frac{10913231}{4790234}a^{15}+\frac{9619740}{2395117}a^{14}+\frac{25987191}{2395117}a^{13}-\frac{37770831}{4790234}a^{12}-\frac{72143582}{2395117}a^{11}-\frac{2147370}{2395117}a^{10}+\frac{102485918}{2395117}a^{9}+\frac{45893771}{2395117}a^{8}-\frac{63840020}{2395117}a^{7}-\frac{95393695}{4790234}a^{6}+\frac{12568371}{2395117}a^{5}+\frac{39501483}{4790234}a^{4}+\frac{354911}{4790234}a^{3}-\frac{28738343}{4790234}a^{2}+\frac{14203207}{4790234}a+\frac{965102}{2395117}$, $\frac{1065032}{2395117}a^{17}-\frac{1821712}{2395117}a^{16}-\frac{47605001}{9580468}a^{15}+\frac{22494823}{4790234}a^{14}+\frac{120215195}{4790234}a^{13}-\frac{39676571}{9580468}a^{12}-\frac{631687615}{9580468}a^{11}-\frac{236862527}{9580468}a^{10}+\frac{824223003}{9580468}a^{9}+\frac{136433728}{2395117}a^{8}-\frac{526285071}{9580468}a^{7}-\frac{361105223}{9580468}a^{6}+\frac{57456566}{2395117}a^{5}+\frac{13951356}{2395117}a^{4}-\frac{78158809}{9580468}a^{3}+\frac{473476}{2395117}a^{2}+\frac{6196131}{9580468}a+\frac{5179739}{9580468}$, $\frac{2063429}{9580468}a^{17}-\frac{1737305}{4790234}a^{16}-\frac{5893664}{2395117}a^{15}+\frac{22974525}{9580468}a^{14}+\frac{120006315}{9580468}a^{13}-\frac{31958097}{9580468}a^{12}-\frac{321653551}{9580468}a^{11}-\frac{31846529}{4790234}a^{10}+\frac{461926701}{9580468}a^{9}+\frac{165834849}{9580468}a^{8}-\frac{203684573}{4790234}a^{7}-\frac{47246115}{4790234}a^{6}+\frac{286268723}{9580468}a^{5}-\frac{1460608}{2395117}a^{4}-\frac{123924291}{9580468}a^{3}+\frac{36548735}{9580468}a^{2}+\frac{6265132}{2395117}a-\frac{6319837}{4790234}$, $\frac{105976}{2395117}a^{17}+\frac{3391845}{9580468}a^{16}-\frac{3458708}{2395117}a^{15}-\frac{9114069}{2395117}a^{14}+\frac{86731523}{9580468}a^{13}+\frac{192543295}{9580468}a^{12}-\frac{194290745}{9580468}a^{11}-\frac{548830509}{9580468}a^{10}+\frac{57881123}{4790234}a^{9}+\frac{793742005}{9580468}a^{8}+\frac{76335333}{9580468}a^{7}-\frac{296463441}{4790234}a^{6}-\frac{10391523}{4790234}a^{5}+\frac{279155253}{9580468}a^{4}-\frac{15594287}{2395117}a^{3}-\frac{51578073}{9580468}a^{2}+\frac{3703289}{9580468}a+\frac{2461893}{4790234}$, $\frac{2459983}{9580468}a^{17}-\frac{105135}{2395117}a^{16}-\frac{8278669}{2395117}a^{15}-\frac{17223225}{9580468}a^{14}+\frac{169087389}{9580468}a^{13}+\frac{192240319}{9580468}a^{12}-\frac{354252129}{9580468}a^{11}-\frac{334200037}{4790234}a^{10}+\frac{173880095}{9580468}a^{9}+\frac{923316093}{9580468}a^{8}+\frac{111734497}{4790234}a^{7}-\frac{266253945}{4790234}a^{6}-\frac{145323135}{9580468}a^{5}+\frac{50406675}{2395117}a^{4}+\frac{288567}{9580468}a^{3}-\frac{51005727}{9580468}a^{2}+\frac{3663614}{2395117}a+\frac{1261203}{2395117}$, $\frac{1053213}{9580468}a^{17}-\frac{2203421}{2395117}a^{16}+\frac{457472}{2395117}a^{15}+\frac{83974919}{9580468}a^{14}-\frac{26197425}{9580468}a^{13}-\frac{363558247}{9580468}a^{12}-\frac{38342651}{9580468}a^{11}+\frac{424156601}{4790234}a^{10}+\frac{414418403}{9580468}a^{9}-\frac{1004356225}{9580468}a^{8}-\frac{167215840}{2395117}a^{7}+\frac{302722467}{4790234}a^{6}+\frac{281971181}{9580468}a^{5}-\frac{76477039}{2395117}a^{4}+\frac{55838341}{9580468}a^{3}+\frac{79432915}{9580468}a^{2}-\frac{1667009}{4790234}a+\frac{1195093}{4790234}$ a, 1226575/9580468a^17 - 910355/9580468a^16 - 13458809/9580468a^15 - 4434301/9580468a^14 + 28898129/4790234a^13 + 79740953/9580468a^12 - 77859779/9580468a^11 - 131575363/4790234a^10 - 27436295/2395117a^9 + 68341827/2395117a^8 + 77488806/2395117a^7 - 43378179/9580468a^6 - 171087015/9580468a^5 + 19493941/9580468a^4 + 13813208/2395117a^3 - 21027625/4790234a^2 - 3150941/4790234a - 5455267/9580468, 370110/2395117a^17 - 1432635/4790234a^16 - 16968629/9580468a^15 + 5358439/2395117a^14 + 44970767/4790234a^13 - 46256023/9580468a^12 - 261971145/9580468a^11 - 11296619/9580468a^10 + 417418267/9580468a^9 + 36884181/2395117a^8 - 356526911/9580468a^7 - 158753111/9580468a^6 + 76578297/4790234a^5 + 10622926/2395117a^4 + 32473/9580468a^3 + 8857955/4790234a^2 - 26838171/9580468a + 8850501/9580468, 3309081/9580468a^17 - 582659/4790234a^16 - 45925259/9580468a^15 - 10537751/9580468a^14 + 242476437/9580468a^13 + 46148803/2395117a^12 - 282230303/4790234a^11 - 721557587/9580468a^10 + 249076477/4790234a^9 + 1057105067/9580468a^8 - 57916219/9580468a^7 - 585856155/9580468a^6 + 19574229/9580468a^5 + 27964485/2395117a^4 - 17834405/2395117a^3 + 29246005/9580468a^2 - 26561221/9580468a - 8631681/9580468, 1957829/4790234a^17 - 6412013/4790234a^16 - 31135395/9580468a^15 + 51728147/4790234a^14 + 34810767/2395117a^13 - 337259045/9580468a^12 - 445973139/9580468a^11 + 571856047/9580468a^10 + 869069969/9580468a^9 - 136937512/2395117a^8 - 907348363/9580468a^7 + 367806235/9580468a^6 + 116915537/2395117a^5 - 59231418/2395117a^4 - 69449905/9580468a^3 + 17686567/4790234a^2 + 10966795/9580468a + 4108429/9580468, 4108429/9580468a^17 - 8024087/9580468a^16 - 40585551/9580468a^15 + 47569111/9580468a^14 + 46033434/2395117a^13 - 52966059/9580468a^12 - 418691891/9580468a^11 - 31274623/2395117a^10 + 192703955/4790234a^9 + 89289713/4790234a^8 - 15074361/2395117a^7 + 134963711/9580468a^6 - 22698199/9580468a^5 - 217047979/9580468a^4 + 9930270/2395117a^3 + 12226940/2395117a^2 - 9469709/4790234a - 1386327/9580468, 3443937/9580468a^17 - 9618875/9580468a^16 - 7259242/2395117a^15 + 69751653/9580468a^14 + 32059634/2395117a^13 - 91539645/4790234a^12 - 88436802/2395117a^11 + 210062003/9580468a^10 + 531219777/9580468a^9 - 63297601/4790234a^8 - 382299877/9580468a^7 + 29687367/2395117a^6 + 115203053/9580468a^5 - 121893461/9580468a^4 + 18809023/9580468a^3 - 3047767/4790234a^2 + 12476411/9580468a + 323963/4790234, 1190897/4790234a^17 - 1454889/2395117a^16 - 10913231/4790234a^15 + 9619740/2395117a^14 + 25987191/2395117a^13 - 37770831/4790234a^12 - 72143582/2395117a^11 - 2147370/2395117a^10 + 102485918/2395117a^9 + 45893771/2395117a^8 - 63840020/2395117a^7 - 95393695/4790234a^6 + 12568371/2395117a^5 + 39501483/4790234a^4 + 354911/4790234a^3 - 28738343/4790234a^2 + 14203207/4790234a + 965102/2395117, 1065032/2395117a^17 - 1821712/2395117a^16 - 47605001/9580468a^15 + 22494823/4790234a^14 + 120215195/4790234a^13 - 39676571/9580468a^12 - 631687615/9580468a^11 - 236862527/9580468a^10 + 824223003/9580468a^9 + 136433728/2395117a^8 - 526285071/9580468a^7 - 361105223/9580468a^6 + 57456566/2395117a^5 + 13951356/2395117a^4 - 78158809/9580468a^3 + 473476/2395117a^2 + 6196131/9580468a + 5179739/9580468, 2063429/9580468a^17 - 1737305/4790234a^16 - 5893664/2395117a^15 + 22974525/9580468a^14 + 120006315/9580468a^13 - 31958097/9580468a^12 - 321653551/9580468a^11 - 31846529/4790234a^10 + 461926701/9580468a^9 + 165834849/9580468a^8 - 203684573/4790234a^7 - 47246115/4790234a^6 + 286268723/9580468a^5 - 1460608/2395117a^4 - 123924291/9580468a^3 + 36548735/9580468a^2 + 6265132/2395117a - 6319837/4790234, 105976/2395117a^17 + 3391845/9580468a^16 - 3458708/2395117a^15 - 9114069/2395117a^14 + 86731523/9580468a^13 + 192543295/9580468a^12 - 194290745/9580468a^11 - 548830509/9580468a^10 + 57881123/4790234a^9 + 793742005/9580468a^8 + 76335333/9580468a^7 - 296463441/4790234a^6 - 10391523/4790234a^5 + 279155253/9580468a^4 - 15594287/2395117a^3 - 51578073/9580468a^2 + 3703289/9580468a + 2461893/4790234, 2459983/9580468a^17 - 105135/2395117a^16 - 8278669/2395117a^15 - 17223225/9580468a^14 + 169087389/9580468a^13 + 192240319/9580468a^12 - 354252129/9580468a^11 - 334200037/4790234a^10 + 173880095/9580468a^9 + 923316093/9580468a^8 + 111734497/4790234a^7 - 266253945/4790234a^6 - 145323135/9580468a^5 + 50406675/2395117a^4 + 288567/9580468a^3 - 51005727/9580468a^2 + 3663614/2395117a + 1261203/2395117, 1053213/9580468a^17 - 2203421/2395117a^16 + 457472/2395117a^15 + 83974919/9580468a^14 - 26197425/9580468a^13 - 363558247/9580468a^12 - 38342651/9580468a^11 + 424156601/4790234a^10 + 414418403/9580468a^9 - 1004356225/9580468a^8 - 167215840/2395117a^7 + 302722467/4790234a^6 + 281971181/9580468a^5 - 76477039/2395117a^4 + 55838341/9580468a^3 + 79432915/9580468a^2 - 1667009/4790234a + 1195093/4790234	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:	$ 144162.848433 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 144162.848433 \cdot 1}{2\cdot\sqrt{385398969258228918652928}}\cr\approx \mathstrut & 0.185305160798 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*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^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*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^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*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^18 - x^17 - 13*x^16 + 4*x^15 + 70*x^14 + 21*x^13 - 184*x^12 - 139*x^11 + 233*x^10 + 255*x^9 - 148*x^8 - 188*x^7 + 84*x^6 + 61*x^5 - 48*x^4 - 5*x^3 + 4*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

$C_2\times S_4^3.S_4$ (as 18T912):

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 solvable group of order 663552

The 330 conjugacy class representatives for $C_2\times S_4^3.S_4$

Character table for $C_2\times S_4^3.S_4$

Intermediate fields

3.3.148.1, 9.9.53038958912.1

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

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 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }^{3}$	${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$	${\href{/padicField/7.9.0.1}{9} }^{2}$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$	${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	R	$18$	${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. 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
$2$ 2	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
$2$ 2	2.9.6.1	$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
$37$ 37	37.3.0.1	$x^{3} + 6 x + 35$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	37.3.0.1	$x^{3} + 6 x + 35$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.8.4.1	$x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$137$ 137	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.1.2	$x^{2} + 411$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	137.4.0.1	$x^{4} + x^{2} + 95 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$16361$ 16361	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{16361}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$

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