LMFDB - Number field 18.0.18294428062468623673667584.4

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736)

gp: K = bnfinit(y^18 - 8*y^17 + 33*y^16 - 114*y^15 + 383*y^14 - 1156*y^13 + 3191*y^12 - 8326*y^11 + 19412*y^10 - 39456*y^9 + 73892*y^8 - 130472*y^7 + 201816*y^6 - 249352*y^5 + 231552*y^4 - 154816*y^3 + 70784*y^2 - 20064*y + 2736, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736)

$ x^{18} - 8 x^{17} + 33 x^{16} - 114 x^{15} + 383 x^{14} - 1156 x^{13} + 3191 x^{12} - 8326 x^{11} + \cdots + 2736 $ x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736

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:	$-18294428062468623673667584$ -18294428062468623673667584 $\medspace = -\,2^{12}\cdot 7^{12}\cdot 19^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$25.32$	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:	$2^{2/3}7^{2/3}19^{1/2}\approx 25.319909997284967$
Ramified primes:	$2$, $7$, $19$ 2, 7, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-19}) $
$\card{ \Gal(K/\Q) }$:	$18$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{5}{12}a^{5}+\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}-\frac{1}{12}a^{6}-\frac{1}{6}a^{5}+\frac{5}{12}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{8}+\frac{1}{12}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a$, $\frac{1}{24}a^{12}-\frac{1}{24}a^{10}-\frac{1}{24}a^{8}+\frac{1}{6}a^{7}+\frac{5}{24}a^{6}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{24}a^{13}-\frac{1}{24}a^{11}-\frac{1}{24}a^{9}-\frac{1}{8}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{144}a^{14}+\frac{1}{72}a^{13}-\frac{1}{48}a^{12}+\frac{1}{36}a^{11}+\frac{1}{48}a^{10}-\frac{1}{72}a^{9}+\frac{11}{144}a^{8}+\frac{1}{36}a^{7}+\frac{7}{36}a^{6}-\frac{7}{36}a^{5}-\frac{13}{36}a^{4}-\frac{1}{36}a^{2}$, $\frac{1}{1440}a^{15}+\frac{1}{360}a^{14}+\frac{5}{288}a^{13}+\frac{1}{90}a^{12}-\frac{49}{1440}a^{11}-\frac{5}{144}a^{10}-\frac{41}{1440}a^{9}-\frac{2}{45}a^{8}-\frac{103}{720}a^{6}-\frac{23}{120}a^{5}+\frac{7}{90}a^{4}+\frac{113}{360}a^{3}+\frac{31}{90}a^{2}+\frac{1}{3}a+\frac{3}{20}$, $\frac{1}{4320}a^{16}-\frac{11}{4320}a^{14}-\frac{1}{1080}a^{13}+\frac{7}{4320}a^{12}+\frac{11}{720}a^{11}-\frac{3}{160}a^{10}+\frac{1}{216}a^{9}-\frac{11}{180}a^{8}+\frac{457}{2160}a^{7}-\frac{43}{180}a^{6}+\frac{11}{45}a^{5}-\frac{17}{40}a^{4}+\frac{83}{270}a^{3}+\frac{53}{135}a^{2}+\frac{29}{180}a-\frac{1}{30}$, $\frac{1}{5841255867840}a^{17}-\frac{61264333}{584125586784}a^{16}+\frac{666851251}{5841255867840}a^{15}-\frac{3689448223}{2920627933920}a^{14}-\frac{116236643443}{5841255867840}a^{13}+\frac{2648915627}{1460313966960}a^{12}+\frac{54238309007}{1947085289280}a^{11}+\frac{5409473597}{584125586784}a^{10}+\frac{60853489577}{2920627933920}a^{9}-\frac{211857668867}{2920627933920}a^{8}-\frac{14143562783}{1460313966960}a^{7}+\frac{90660966097}{486771322320}a^{6}-\frac{53075472473}{162257107440}a^{5}-\frac{30236583101}{730156983480}a^{4}+\frac{71699295749}{146031396696}a^{3}+\frac{268725277531}{730156983480}a^{2}+\frac{596899219}{3202442910}a-\frac{490221533}{2134961940}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3, 1/2*a^7 - 1/2*a^3, 1/6*a^8 - 1/6*a^7 + 1/6*a^6 - 1/6*a^5 - 1/3*a^4 + 1/3*a^3 - 1/3*a^2 + 1/3*a, 1/12*a^9 - 1/12*a^8 - 1/6*a^7 + 1/6*a^6 - 5/12*a^5 + 5/12*a^4 + 1/3*a^3 + 1/6*a^2 - 1/2*a, 1/12*a^10 - 1/12*a^8 - 1/6*a^7 - 1/12*a^6 - 1/6*a^5 + 5/12*a^4 - 1/6*a^3 + 1/3*a^2 - 1/6*a, 1/12*a^11 - 1/12*a^8 + 1/12*a^7 + 1/6*a^6 - 1/6*a^5 - 1/12*a^4 - 1/2*a^3 - 1/3*a^2 - 1/6*a, 1/24*a^12 - 1/24*a^10 - 1/24*a^8 + 1/6*a^7 + 5/24*a^6 - 1/3*a^5 + 1/6*a^4 - 1/3*a^3 - 1/2*a^2 - 1/3*a - 1/2, 1/24*a^13 - 1/24*a^11 - 1/24*a^9 - 1/8*a^7 - 1/6*a^5 - 1/2*a^4 + 1/6*a^3 + 1/6*a, 1/144*a^14 + 1/72*a^13 - 1/48*a^12 + 1/36*a^11 + 1/48*a^10 - 1/72*a^9 + 11/144*a^8 + 1/36*a^7 + 7/36*a^6 - 7/36*a^5 - 13/36*a^4 - 1/36*a^2, 1/1440*a^15 + 1/360*a^14 + 5/288*a^13 + 1/90*a^12 - 49/1440*a^11 - 5/144*a^10 - 41/1440*a^9 - 2/45*a^8 - 103/720*a^6 - 23/120*a^5 + 7/90*a^4 + 113/360*a^3 + 31/90*a^2 + 1/3*a + 3/20, 1/4320*a^16 - 11/4320*a^14 - 1/1080*a^13 + 7/4320*a^12 + 11/720*a^11 - 3/160*a^10 + 1/216*a^9 - 11/180*a^8 + 457/2160*a^7 - 43/180*a^6 + 11/45*a^5 - 17/40*a^4 + 83/270*a^3 + 53/135*a^2 + 29/180*a - 1/30, 1/5841255867840*a^17 - 61264333/584125586784*a^16 + 666851251/5841255867840*a^15 - 3689448223/2920627933920*a^14 - 116236643443/5841255867840*a^13 + 2648915627/1460313966960*a^12 + 54238309007/1947085289280*a^11 + 5409473597/584125586784*a^10 + 60853489577/2920627933920*a^9 - 211857668867/2920627933920*a^8 - 14143562783/1460313966960*a^7 + 90660966097/486771322320*a^6 - 53075472473/162257107440*a^5 - 30236583101/730156983480*a^4 + 71699295749/146031396696*a^3 + 268725277531/730156983480*a^2 + 596899219/3202442910*a - 490221533/2134961940

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{3}$, which has order $3$

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:	$\frac{11093517673}{1947085289280}a^{17}-\frac{31837538681}{973542644640}a^{16}+\frac{183307987619}{1947085289280}a^{15}-\frac{291733809431}{973542644640}a^{14}+\frac{659509070143}{649028429760}a^{13}-\frac{144221894429}{54085702480}a^{12}+\frac{13040231010797}{1947085289280}a^{11}-\frac{1792620978951}{108171404960}a^{10}+\frac{30351912636253}{973542644640}a^{9}-\frac{45673482685579}{973542644640}a^{8}+\frac{38529028885909}{486771322320}a^{7}-\frac{11743798210981}{97354264464}a^{6}+\frac{11772887448029}{162257107440}a^{5}+\frac{1967369994013}{16225710744}a^{4}-\frac{73729101211723}{243385661160}a^{3}+\frac{23286365724577}{81128553720}a^{2}-\frac{144679520551}{1067480970}a+\frac{3844336747}{142330796}$, $\frac{11093517673}{1947085289280}a^{17}-\frac{31837538681}{973542644640}a^{16}+\frac{183307987619}{1947085289280}a^{15}-\frac{291733809431}{973542644640}a^{14}+\frac{659509070143}{649028429760}a^{13}-\frac{144221894429}{54085702480}a^{12}+\frac{13040231010797}{1947085289280}a^{11}-\frac{1792620978951}{108171404960}a^{10}+\frac{30351912636253}{973542644640}a^{9}-\frac{45673482685579}{973542644640}a^{8}+\frac{38529028885909}{486771322320}a^{7}-\frac{11743798210981}{97354264464}a^{6}+\frac{11772887448029}{162257107440}a^{5}+\frac{1967369994013}{16225710744}a^{4}-\frac{73729101211723}{243385661160}a^{3}+\frac{23286365724577}{81128553720}a^{2}-\frac{144679520551}{1067480970}a+\frac{3986667543}{142330796}$, $\frac{88014253}{4921024320}a^{17}-\frac{143858669}{1230256080}a^{16}+\frac{2041870939}{4921024320}a^{15}-\frac{578445509}{410085360}a^{14}+\frac{1549916611}{328068288}a^{13}-\frac{33328729411}{2460512160}a^{12}+\frac{59927738651}{1640341440}a^{11}-\frac{57561619343}{615128040}a^{10}+\frac{505026927643}{2460512160}a^{9}-\frac{966743645921}{2460512160}a^{8}+\frac{22221808811}{30756402}a^{7}-\frac{506045350691}{410085360}a^{6}+\frac{707736533317}{410085360}a^{5}-\frac{557314701691}{307564020}a^{4}+\frac{92965791587}{68347560}a^{3}-\frac{430649941447}{615128040}a^{2}+\frac{1217956441}{5395860}a-\frac{65518399}{1798620}$, $\frac{13001}{4177440}a^{17}-\frac{94909}{1392480}a^{16}+\frac{1672013}{4177440}a^{15}-\frac{5925641}{4177440}a^{14}+\frac{3945247}{835488}a^{13}-\frac{21533563}{1392480}a^{12}+\frac{61505047}{1392480}a^{11}-\frac{489851843}{4177440}a^{10}+\frac{102431291}{348120}a^{9}-\frac{664851641}{1044360}a^{8}+\frac{166724465}{139248}a^{7}-\frac{375848291}{174060}a^{6}+\frac{1264468399}{348120}a^{5}-\frac{5132546513}{1044360}a^{4}+\frac{1245319163}{261090}a^{3}-\frac{89259493}{29010}a^{2}+\frac{69123527}{58020}a-\frac{2079469}{9670}$, $\frac{27438292973}{730156983480}a^{17}-\frac{124900799413}{486771322320}a^{16}+\frac{1340988936043}{1460313966960}a^{15}-\frac{1118631497911}{365078491740}a^{14}+\frac{15030453322513}{1460313966960}a^{13}-\frac{3613404389249}{121692830580}a^{12}+\frac{38668374014893}{486771322320}a^{11}-\frac{74279142170599}{365078491740}a^{10}+\frac{72623176664753}{162257107440}a^{9}-\frac{12\!\cdots\!81}{1460313966960}a^{8}+\frac{375009000346847}{243385661160}a^{7}-\frac{32143726933559}{12169283058}a^{6}+\frac{445623839164393}{121692830580}a^{5}-\frac{66877247361923}{18253924587}a^{4}+\frac{904217354685913}{365078491740}a^{3}-\frac{128282315404297}{121692830580}a^{2}+\frac{264036897109}{1067480970}a-\frac{716960303}{35582699}$, $\frac{126861892847}{5841255867840}a^{17}-\frac{347320647167}{2920627933920}a^{16}+\frac{1945272008699}{5841255867840}a^{15}-\frac{209048756003}{194708528928}a^{14}+\frac{1411897864367}{389417057856}a^{13}-\frac{2730088805807}{292062793392}a^{12}+\frac{1017165281203}{43268561984}a^{11}-\frac{168787901360023}{2920627933920}a^{10}+\frac{19292835734663}{182539245870}a^{9}-\frac{90736622480609}{584125586784}a^{8}+\frac{76719960986963}{292062793392}a^{7}-\frac{15627479509669}{40564276860}a^{6}+\frac{81746326810469}{486771322320}a^{5}+\frac{398065681952999}{730156983480}a^{4}-\frac{141847893410981}{121692830580}a^{3}+\frac{781395420644533}{730156983480}a^{2}-\frac{814575975694}{1601221455}a+\frac{58444915408}{533740485}$, $\frac{1247535713}{162257107440}a^{17}-\frac{25198438601}{486771322320}a^{16}+\frac{17956116233}{97354264464}a^{15}-\frac{299802256939}{486771322320}a^{14}+\frac{335130914579}{162257107440}a^{13}-\frac{2893233700279}{486771322320}a^{12}+\frac{7739435895419}{486771322320}a^{11}-\frac{19787368666229}{486771322320}a^{10}+\frac{120520361431}{1352142562}a^{9}-\frac{10238812154537}{60846415290}a^{8}+\frac{18594368048191}{60846415290}a^{7}-\frac{63533710128617}{121692830580}a^{6}+\frac{7298635049942}{10141069215}a^{5}-\frac{43259861418883}{60846415290}a^{4}+\frac{14127333782689}{30423207645}a^{3}-\frac{10917653243171}{60846415290}a^{2}+\frac{10809760451}{355826990}a+\frac{199583758}{177913495}$, $\frac{5848777097}{1168251173568}a^{17}-\frac{4865887687}{584125586784}a^{16}-\frac{47435261269}{1168251173568}a^{15}+\frac{85314894985}{584125586784}a^{14}-\frac{577125759515}{1168251173568}a^{13}+\frac{670320884497}{292062793392}a^{12}-\frac{2748579755833}{389417057856}a^{11}+\frac{11669775641681}{584125586784}a^{10}-\frac{35172976952207}{584125586784}a^{9}+\frac{84215903977679}{584125586784}a^{8}-\frac{79755797897395}{292062793392}a^{7}+\frac{16852391064635}{32451421488}a^{6}-\frac{95546627767651}{97354264464}a^{5}+\frac{213404326350821}{146031396696}a^{4}-\frac{218983431218551}{146031396696}a^{3}+\frac{144086176094465}{146031396696}a^{2}-\frac{121274337974}{320244291}a+\frac{27673339367}{426992388}$ 11093517673/1947085289280a^17 - 31837538681/973542644640a^16 + 183307987619/1947085289280a^15 - 291733809431/973542644640a^14 + 659509070143/649028429760a^13 - 144221894429/54085702480a^12 + 13040231010797/1947085289280a^11 - 1792620978951/108171404960a^10 + 30351912636253/973542644640a^9 - 45673482685579/973542644640a^8 + 38529028885909/486771322320a^7 - 11743798210981/97354264464a^6 + 11772887448029/162257107440a^5 + 1967369994013/16225710744a^4 - 73729101211723/243385661160a^3 + 23286365724577/81128553720a^2 - 144679520551/1067480970a + 3844336747/142330796, 11093517673/1947085289280a^17 - 31837538681/973542644640a^16 + 183307987619/1947085289280a^15 - 291733809431/973542644640a^14 + 659509070143/649028429760a^13 - 144221894429/54085702480a^12 + 13040231010797/1947085289280a^11 - 1792620978951/108171404960a^10 + 30351912636253/973542644640a^9 - 45673482685579/973542644640a^8 + 38529028885909/486771322320a^7 - 11743798210981/97354264464a^6 + 11772887448029/162257107440a^5 + 1967369994013/16225710744a^4 - 73729101211723/243385661160a^3 + 23286365724577/81128553720a^2 - 144679520551/1067480970a + 3986667543/142330796, 88014253/4921024320a^17 - 143858669/1230256080a^16 + 2041870939/4921024320a^15 - 578445509/410085360a^14 + 1549916611/328068288a^13 - 33328729411/2460512160a^12 + 59927738651/1640341440a^11 - 57561619343/615128040a^10 + 505026927643/2460512160a^9 - 966743645921/2460512160a^8 + 22221808811/30756402a^7 - 506045350691/410085360a^6 + 707736533317/410085360a^5 - 557314701691/307564020a^4 + 92965791587/68347560a^3 - 430649941447/615128040a^2 + 1217956441/5395860a - 65518399/1798620, 13001/4177440a^17 - 94909/1392480a^16 + 1672013/4177440a^15 - 5925641/4177440a^14 + 3945247/835488a^13 - 21533563/1392480a^12 + 61505047/1392480a^11 - 489851843/4177440a^10 + 102431291/348120a^9 - 664851641/1044360a^8 + 166724465/139248a^7 - 375848291/174060a^6 + 1264468399/348120a^5 - 5132546513/1044360a^4 + 1245319163/261090a^3 - 89259493/29010a^2 + 69123527/58020a - 2079469/9670, 27438292973/730156983480a^17 - 124900799413/486771322320a^16 + 1340988936043/1460313966960a^15 - 1118631497911/365078491740a^14 + 15030453322513/1460313966960a^13 - 3613404389249/121692830580a^12 + 38668374014893/486771322320a^11 - 74279142170599/365078491740a^10 + 72623176664753/162257107440a^9 - 1236839170916581/1460313966960a^8 + 375009000346847/243385661160a^7 - 32143726933559/12169283058a^6 + 445623839164393/121692830580a^5 - 66877247361923/18253924587a^4 + 904217354685913/365078491740a^3 - 128282315404297/121692830580a^2 + 264036897109/1067480970a - 716960303/35582699, 126861892847/5841255867840a^17 - 347320647167/2920627933920a^16 + 1945272008699/5841255867840a^15 - 209048756003/194708528928a^14 + 1411897864367/389417057856a^13 - 2730088805807/292062793392a^12 + 1017165281203/43268561984a^11 - 168787901360023/2920627933920a^10 + 19292835734663/182539245870a^9 - 90736622480609/584125586784a^8 + 76719960986963/292062793392a^7 - 15627479509669/40564276860a^6 + 81746326810469/486771322320a^5 + 398065681952999/730156983480a^4 - 141847893410981/121692830580a^3 + 781395420644533/730156983480a^2 - 814575975694/1601221455a + 58444915408/533740485, 1247535713/162257107440a^17 - 25198438601/486771322320a^16 + 17956116233/97354264464a^15 - 299802256939/486771322320a^14 + 335130914579/162257107440a^13 - 2893233700279/486771322320a^12 + 7739435895419/486771322320a^11 - 19787368666229/486771322320a^10 + 120520361431/1352142562a^9 - 10238812154537/60846415290a^8 + 18594368048191/60846415290a^7 - 63533710128617/121692830580a^6 + 7298635049942/10141069215a^5 - 43259861418883/60846415290a^4 + 14127333782689/30423207645a^3 - 10917653243171/60846415290a^2 + 10809760451/355826990a + 199583758/177913495, 5848777097/1168251173568a^17 - 4865887687/584125586784a^16 - 47435261269/1168251173568a^15 + 85314894985/584125586784a^14 - 577125759515/1168251173568a^13 + 670320884497/292062793392a^12 - 2748579755833/389417057856a^11 + 11669775641681/584125586784a^10 - 35172976952207/584125586784a^9 + 84215903977679/584125586784a^8 - 79755797897395/292062793392a^7 + 16852391064635/32451421488a^6 - 95546627767651/97354264464a^5 + 213404326350821/146031396696a^4 - 218983431218551/146031396696a^3 + 144086176094465/146031396696a^2 - 121274337974/320244291a + 27673339367/426992388	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:	$ 229812.209675 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 229812.209675 \cdot 3}{2\cdot\sqrt{18294428062468623673667584}}\cr\approx \mathstrut & 1.23005280159 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736)

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 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736, 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 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736);

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 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736);

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{-19}) $, 3.1.3724.2 x3, 3.1.76.1 x3, 3.1.931.1 x3, 3.1.3724.1 x3, 6.0.263495344.2, 6.0.109744.2, 6.0.16468459.2, 6.0.263495344.1, 9.1.981256661056.1 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:	9.1.981256661056.1
Minimal sibling:	9.1.981256661056.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.2.0.1}{2} }^{9}$	${\href{/padicField/5.3.0.1}{3} }^{6}$	R	${\href{/padicField/11.3.0.1}{3} }^{6}$	${\href{/padicField/13.2.0.1}{2} }^{9}$	${\href{/padicField/17.3.0.1}{3} }^{6}$	R	${\href{/padicField/23.3.0.1}{3} }^{6}$	${\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.2.0.1}{2} }^{9}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.3.0.1}{3} }^{6}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

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	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$19$ 19	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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