LMFDB - Number field 18.0.5080969518089676267073536.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51)

gp: K = bnfinit(y^18 - 9*y^17 + 39*y^16 - 108*y^15 + 240*y^14 - 504*y^13 + 988*y^12 - 1638*y^11 + 2775*y^10 - 5295*y^9 + 8883*y^8 - 11094*y^7 + 10882*y^6 - 9186*y^5 + 6900*y^4 - 4230*y^3 + 1815*y^2 - 459*y + 51, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51)

$ x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 240 x^{14} - 504 x^{13} + 988 x^{12} - 1638 x^{11} + \cdots + 51 $ x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51

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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-5080969518089676267073536$ -5080969518089676267073536 $\medspace = -\,2^{12}\cdot 3^{21}\cdot 17^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$23.58$	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:	$2^{2/3}3^{7/6}17^{1/2}\approx 23.58047712423637$
Ramified primes:	$2$, $3$, $17$ 2, 3, 17	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-51}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{408}a^{14}-\frac{7}{408}a^{13}-\frac{1}{204}a^{12}+\frac{1}{408}a^{11}+\frac{47}{408}a^{10}+\frac{41}{204}a^{9}+\frac{43}{408}a^{8}+\frac{41}{408}a^{7}+\frac{67}{408}a^{6}+\frac{11}{68}a^{5}+\frac{29}{136}a^{4}-\frac{13}{136}a^{3}+\frac{29}{68}a^{2}-\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{408}a^{15}-\frac{1}{8}a^{13}-\frac{13}{408}a^{12}-\frac{2}{17}a^{11}+\frac{1}{136}a^{10}+\frac{5}{408}a^{9}+\frac{3}{34}a^{8}+\frac{2}{17}a^{7}-\frac{77}{408}a^{6}+\frac{47}{136}a^{5}-\frac{6}{17}a^{4}+\frac{1}{136}a^{3}-\frac{19}{136}a^{2}+\frac{3}{8}$, $\frac{1}{105142824}a^{16}-\frac{1}{13142853}a^{15}-\frac{42307}{35047608}a^{14}+\frac{888587}{105142824}a^{13}-\frac{3129253}{26285706}a^{12}+\frac{3659555}{35047608}a^{11}-\frac{5383447}{105142824}a^{10}-\frac{4366037}{52571412}a^{9}+\frac{2570183}{17523804}a^{8}+\frac{22595113}{105142824}a^{7}+\frac{3325021}{105142824}a^{6}+\frac{318311}{17523804}a^{5}-\frac{792827}{11682536}a^{4}+\frac{1909421}{11682536}a^{3}+\frac{5182351}{17523804}a^{2}+\frac{22543}{66504}a-\frac{453541}{1030812}$, $\frac{1}{5362284024}a^{17}+\frac{1}{315428472}a^{16}-\frac{1157933}{5362284024}a^{15}+\frac{1361663}{1340571006}a^{14}+\frac{507579859}{5362284024}a^{13}-\frac{251436847}{5362284024}a^{12}-\frac{86567585}{2681142012}a^{11}+\frac{558664723}{5362284024}a^{10}-\frac{36756923}{2681142012}a^{9}+\frac{361736023}{5362284024}a^{8}+\frac{509849251}{2681142012}a^{7}-\frac{868465709}{5362284024}a^{6}-\frac{224065097}{1787428008}a^{5}-\frac{3844219}{99301556}a^{4}-\frac{867963415}{1787428008}a^{3}+\frac{677653045}{1787428008}a^{2}-\frac{43738759}{105142824}a-\frac{18554209}{52571412}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^8 - 1/2*a^4 - 1/2, 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/4*a^6 + 1/4*a^5 + 1/4*a^4 - 1/2*a^3 - 1/4*a^2 + 1/4*a - 1/4, 1/4*a^11 - 1/4*a^8 - 1/4*a^7 - 1/4*a^4 + 1/4*a^3 - 1/2*a - 1/4, 1/4*a^12 - 1/4*a^9 - 1/4*a^8 - 1/4*a^5 + 1/4*a^4 - 1/2*a^2 - 1/4*a, 1/4*a^13 - 1/4*a^8 - 1/2*a^5 + 1/4*a^4 - 1/2*a^3 - 1/2*a^2 + 1/4*a + 1/4, 1/408*a^14 - 7/408*a^13 - 1/204*a^12 + 1/408*a^11 + 47/408*a^10 + 41/204*a^9 + 43/408*a^8 + 41/408*a^7 + 67/408*a^6 + 11/68*a^5 + 29/136*a^4 - 13/136*a^3 + 29/68*a^2 - 1/8*a + 3/8, 1/408*a^15 - 1/8*a^13 - 13/408*a^12 - 2/17*a^11 + 1/136*a^10 + 5/408*a^9 + 3/34*a^8 + 2/17*a^7 - 77/408*a^6 + 47/136*a^5 - 6/17*a^4 + 1/136*a^3 - 19/136*a^2 + 3/8, 1/105142824*a^16 - 1/13142853*a^15 - 42307/35047608*a^14 + 888587/105142824*a^13 - 3129253/26285706*a^12 + 3659555/35047608*a^11 - 5383447/105142824*a^10 - 4366037/52571412*a^9 + 2570183/17523804*a^8 + 22595113/105142824*a^7 + 3325021/105142824*a^6 + 318311/17523804*a^5 - 792827/11682536*a^4 + 1909421/11682536*a^3 + 5182351/17523804*a^2 + 22543/66504*a - 453541/1030812, 1/5362284024*a^17 + 1/315428472*a^16 - 1157933/5362284024*a^15 + 1361663/1340571006*a^14 + 507579859/5362284024*a^13 - 251436847/5362284024*a^12 - 86567585/2681142012*a^11 + 558664723/5362284024*a^10 - 36756923/2681142012*a^9 + 361736023/5362284024*a^8 + 509849251/2681142012*a^7 - 868465709/5362284024*a^6 - 224065097/1787428008*a^5 - 3844219/99301556*a^4 - 867963415/1787428008*a^3 + 677653045/1787428008*a^2 - 43738759/105142824*a - 18554209/52571412

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{6}$, which has order $6$

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:	$8$	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:	$\frac{422561653}{5362284024}a^{17}-\frac{110480143}{157714236}a^{16}+\frac{4016414059}{1340571006}a^{15}-\frac{43849760779}{5362284024}a^{14}+\frac{96439030885}{5362284024}a^{13}-\frac{25219922834}{670285503}a^{12}+\frac{393813958835}{5362284024}a^{11}-\frac{645842540123}{5362284024}a^{10}+\frac{1095379663325}{5362284024}a^{9}-\frac{2104952120027}{5362284024}a^{8}+\frac{3504188060591}{5362284024}a^{7}-\frac{4271994990311}{5362284024}a^{6}+\frac{169334561759}{223428501}a^{5}-\frac{123529800991}{198603112}a^{4}+\frac{818801318861}{1787428008}a^{3}-\frac{120685171973}{446857002}a^{2}+\frac{1341039457}{13142853}a-\frac{1864336529}{105142824}$, $\frac{6553465}{16448724}a^{17}-\frac{6363001}{1935144}a^{16}+\frac{214927765}{16448724}a^{15}-\frac{1090446599}{32897448}a^{14}+\frac{2315995127}{32897448}a^{13}-\frac{604930789}{4112181}a^{12}+\frac{9258860119}{32897448}a^{11}-\frac{14394432619}{32897448}a^{10}+\frac{6330407809}{8224362}a^{9}-\frac{12510780799}{8224362}a^{8}+\frac{78235556611}{32897448}a^{7}-\frac{85341619159}{32897448}a^{6}+\frac{12746600371}{5482908}a^{5}-\frac{2255185845}{1218424}a^{4}+\frac{14301075103}{10965816}a^{3}-\frac{1818884713}{2741454}a^{2}+\frac{126159607}{645048}a-\frac{1984418}{80631}$, $\frac{739159}{8224362}a^{17}-\frac{739159}{967572}a^{16}+\frac{51409469}{16448724}a^{15}-\frac{268513915}{32897448}a^{14}+\frac{577570783}{32897448}a^{13}-\frac{301596919}{8224362}a^{12}+\frac{2327555375}{32897448}a^{11}-\frac{3698983013}{32897448}a^{10}+\frac{3195806995}{16448724}a^{9}-\frac{12507915263}{32897448}a^{8}+\frac{20096927327}{32897448}a^{7}-\frac{22953183221}{32897448}a^{6}+\frac{3494322539}{5482908}a^{5}-\frac{623792335}{1218424}a^{4}+\frac{4007594975}{10965816}a^{3}-\frac{1085445931}{5482908}a^{2}+\frac{39511769}{645048}a-\frac{5107625}{645048}$, $\frac{44442733}{157714236}a^{17}-\frac{391780099}{157714236}a^{16}+\frac{1657300819}{157714236}a^{15}-\frac{8927768777}{315428472}a^{14}+\frac{19448031701}{315428472}a^{13}-\frac{10153990247}{78857118}a^{12}+\frac{79077353023}{315428472}a^{11}-\frac{128403053467}{315428472}a^{10}+\frac{54502662295}{78857118}a^{9}-\frac{422816730583}{315428472}a^{8}+\frac{698163920551}{315428472}a^{7}-\frac{830473919311}{315428472}a^{6}+\frac{128781035071}{52571412}a^{5}-\frac{23345953019}{11682536}a^{4}+\frac{153194467531}{105142824}a^{3}-\frac{43710974741}{52571412}a^{2}+\frac{1823994457}{6184872}a-\frac{284963929}{6184872}$, $\frac{42207043}{297904668}a^{17}-\frac{39445861}{35047608}a^{16}+\frac{424436553}{99301556}a^{15}-\frac{6122062043}{595809336}a^{14}+\frac{12597380615}{595809336}a^{13}-\frac{2188193491}{49650778}a^{12}+\frac{49414691467}{595809336}a^{11}-\frac{73036627687}{595809336}a^{10}+\frac{11021900707}{49650778}a^{9}-\frac{67317446359}{148952334}a^{8}+\frac{398226185131}{595809336}a^{7}-\frac{127115652261}{198603112}a^{6}+\frac{51441998737}{99301556}a^{5}-\frac{77674051165}{198603112}a^{4}+\frac{50031590295}{198603112}a^{3}-\frac{2175187267}{24825389}a^{2}-\frac{76193837}{11682536}a+\frac{29409503}{2920634}$, $\frac{29685943}{297904668}a^{17}-\frac{15221639}{17523804}a^{16}+\frac{360890879}{99301556}a^{15}-\frac{5772326975}{595809336}a^{14}+\frac{12521366651}{595809336}a^{13}-\frac{4362028205}{99301556}a^{12}+\frac{50834498107}{595809336}a^{11}-\frac{81977298247}{595809336}a^{10}+\frac{11662348749}{49650778}a^{9}-\frac{272586173467}{595809336}a^{8}+\frac{445860665275}{595809336}a^{7}-\frac{174457072833}{198603112}a^{6}+\frac{81056668539}{99301556}a^{5}-\frac{132868925515}{198603112}a^{4}+\frac{96198584215}{198603112}a^{3}-\frac{6797970178}{24825389}a^{2}+\frac{1158059131}{11682536}a-\frac{188805787}{11682536}$, $\frac{13584320}{74476167}a^{17}-\frac{170246303}{105142824}a^{16}+\frac{3081161165}{446857002}a^{15}-\frac{359281643}{19219656}a^{14}+\frac{73032351235}{1787428008}a^{13}-\frac{76291284199}{893714004}a^{12}+\frac{99187632535}{595809336}a^{11}-\frac{485355799841}{1787428008}a^{10}+\frac{3314634155}{7207371}a^{9}-\frac{265002314227}{297904668}a^{8}+\frac{2638027113269}{1787428008}a^{7}-\frac{3167721395893}{1787428008}a^{6}+\frac{247191250679}{148952334}a^{5}-\frac{269826544587}{198603112}a^{4}+\frac{197358518125}{198603112}a^{3}-\frac{170661661765}{297904668}a^{2}+\frac{7276886591}{35047608}a-\frac{293152093}{8761902}$, $\frac{263566435}{2681142012}a^{17}-\frac{251155411}{315428472}a^{16}+\frac{4161603899}{1340571006}a^{15}-\frac{41401170347}{5362284024}a^{14}+\frac{87162385745}{5362284024}a^{13}-\frac{91075194073}{2681142012}a^{12}+\frac{11170458529}{172976904}a^{11}-\frac{529592444521}{5362284024}a^{10}+\frac{472125946583}{2681142012}a^{9}-\frac{939238115429}{2681142012}a^{8}+\frac{2878105292191}{5362284024}a^{7}-\frac{3028627319989}{5362284024}a^{6}+\frac{222480253961}{446857002}a^{5}-\frac{77660685181}{198603112}a^{4}+\frac{482091017413}{1787428008}a^{3}-\frac{115110368615}{893714004}a^{2}+\frac{3525138655}{105142824}a-\frac{45296525}{13142853}$ 422561653/5362284024a^17 - 110480143/157714236a^16 + 4016414059/1340571006a^15 - 43849760779/5362284024a^14 + 96439030885/5362284024a^13 - 25219922834/670285503a^12 + 393813958835/5362284024a^11 - 645842540123/5362284024a^10 + 1095379663325/5362284024a^9 - 2104952120027/5362284024a^8 + 3504188060591/5362284024a^7 - 4271994990311/5362284024a^6 + 169334561759/223428501a^5 - 123529800991/198603112a^4 + 818801318861/1787428008a^3 - 120685171973/446857002a^2 + 1341039457/13142853a - 1864336529/105142824, 6553465/16448724a^17 - 6363001/1935144a^16 + 214927765/16448724a^15 - 1090446599/32897448a^14 + 2315995127/32897448a^13 - 604930789/4112181a^12 + 9258860119/32897448a^11 - 14394432619/32897448a^10 + 6330407809/8224362a^9 - 12510780799/8224362a^8 + 78235556611/32897448a^7 - 85341619159/32897448a^6 + 12746600371/5482908a^5 - 2255185845/1218424a^4 + 14301075103/10965816a^3 - 1818884713/2741454a^2 + 126159607/645048a - 1984418/80631, 739159/8224362a^17 - 739159/967572a^16 + 51409469/16448724a^15 - 268513915/32897448a^14 + 577570783/32897448a^13 - 301596919/8224362a^12 + 2327555375/32897448a^11 - 3698983013/32897448a^10 + 3195806995/16448724a^9 - 12507915263/32897448a^8 + 20096927327/32897448a^7 - 22953183221/32897448a^6 + 3494322539/5482908a^5 - 623792335/1218424a^4 + 4007594975/10965816a^3 - 1085445931/5482908a^2 + 39511769/645048a - 5107625/645048, 44442733/157714236a^17 - 391780099/157714236a^16 + 1657300819/157714236a^15 - 8927768777/315428472a^14 + 19448031701/315428472a^13 - 10153990247/78857118a^12 + 79077353023/315428472a^11 - 128403053467/315428472a^10 + 54502662295/78857118a^9 - 422816730583/315428472a^8 + 698163920551/315428472a^7 - 830473919311/315428472a^6 + 128781035071/52571412a^5 - 23345953019/11682536a^4 + 153194467531/105142824a^3 - 43710974741/52571412a^2 + 1823994457/6184872a - 284963929/6184872, 42207043/297904668a^17 - 39445861/35047608a^16 + 424436553/99301556a^15 - 6122062043/595809336a^14 + 12597380615/595809336a^13 - 2188193491/49650778a^12 + 49414691467/595809336a^11 - 73036627687/595809336a^10 + 11021900707/49650778a^9 - 67317446359/148952334a^8 + 398226185131/595809336a^7 - 127115652261/198603112a^6 + 51441998737/99301556a^5 - 77674051165/198603112a^4 + 50031590295/198603112a^3 - 2175187267/24825389a^2 - 76193837/11682536a + 29409503/2920634, 29685943/297904668a^17 - 15221639/17523804a^16 + 360890879/99301556a^15 - 5772326975/595809336a^14 + 12521366651/595809336a^13 - 4362028205/99301556a^12 + 50834498107/595809336a^11 - 81977298247/595809336a^10 + 11662348749/49650778a^9 - 272586173467/595809336a^8 + 445860665275/595809336a^7 - 174457072833/198603112a^6 + 81056668539/99301556a^5 - 132868925515/198603112a^4 + 96198584215/198603112a^3 - 6797970178/24825389a^2 + 1158059131/11682536a - 188805787/11682536, 13584320/74476167a^17 - 170246303/105142824a^16 + 3081161165/446857002a^15 - 359281643/19219656a^14 + 73032351235/1787428008a^13 - 76291284199/893714004a^12 + 99187632535/595809336a^11 - 485355799841/1787428008a^10 + 3314634155/7207371a^9 - 265002314227/297904668a^8 + 2638027113269/1787428008a^7 - 3167721395893/1787428008a^6 + 247191250679/148952334a^5 - 269826544587/198603112a^4 + 197358518125/198603112a^3 - 170661661765/297904668a^2 + 7276886591/35047608a - 293152093/8761902, 263566435/2681142012a^17 - 251155411/315428472a^16 + 4161603899/1340571006a^15 - 41401170347/5362284024a^14 + 87162385745/5362284024a^13 - 91075194073/2681142012a^12 + 11170458529/172976904a^11 - 529592444521/5362284024a^10 + 472125946583/2681142012a^9 - 939238115429/2681142012a^8 + 2878105292191/5362284024a^7 - 3028627319989/5362284024a^6 + 222480253961/446857002a^5 - 77660685181/198603112a^4 + 482091017413/1787428008a^3 - 115110368615/893714004a^2 + 3525138655/105142824a - 45296525/13142853	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:	$ 88275.1610738 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 88275.1610738 \cdot 6}{2\cdot\sqrt{5080969518089676267073536}}\cr\approx \mathstrut & 1.79310323419 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51)

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 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51, 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 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51);

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 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51);

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

$C_3:S_3$ (as 18T4):

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 18

The 6 conjugacy class representatives for $C_3^2 : C_2$

Character table for $C_3^2 : C_2$

Intermediate fields

$\Q(\sqrt{-51}) $, 3.1.1836.2 x3, 3.1.204.1 x3, 3.1.1836.1 x3, 3.1.459.1 x3, 6.0.171915696.1, 6.0.2122416.1, 6.0.171915696.2, 6.0.10744731.1, 9.1.315637217856.1 x9

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 9 sibling:	9.1.315637217856.1
Minimal sibling:	9.1.315637217856.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.3.0.1}{3} }^{6}$	${\href{/padicField/7.2.0.1}{2} }^{9}$	${\href{/padicField/11.3.0.1}{3} }^{6}$	${\href{/padicField/13.3.0.1}{3} }^{6}$	R	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.3.0.1}{3} }^{6}$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.2.0.1}{2} }^{9}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.3.0.1}{3} }^{6}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

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	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$ 3	3.6.7.1	$x^{6} + 6 x^{2} + 6$	$6$	$1$	$7$	$S_3$	$[3/2]_{2}$
	3.6.7.1	$x^{6} + 6 x^{2} + 6$	$6$	$1$	$7$	$S_3$	$[3/2]_{2}$
	3.6.7.1	$x^{6} + 6 x^{2} + 6$	$6$	$1$	$7$	$S_3$	$[3/2]_{2}$
$17$ 17	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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