LMFDB - Number field 23.1.8316624386164136054632388935703.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1)

gp: K = bnfinit(y^23 - 2*y^22 - 3*y^21 + 13*y^20 - 19*y^19 + 21*y^18 - 75*y^16 + 202*y^15 - 378*y^14 + 596*y^13 - 798*y^12 + 955*y^11 - 995*y^10 + 849*y^9 - 610*y^8 + 384*y^7 - 151*y^6 + 2*y^5 + 46*y^4 - 37*y^3 + 3*y^2 + 20*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1)

$ x^{23} - 2 x^{22} - 3 x^{21} + 13 x^{20} - 19 x^{19} + 21 x^{18} - 75 x^{16} + 202 x^{15} - 378 x^{14} + \cdots + 1 $ x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$23$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 11]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-8316624386164136054632388935703$ -8316624386164136054632388935703 $\medspace = -\,647^{11}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$22.10$	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:	$647^{1/2}\approx 25.436194683953808$
Ramified primes:	$647$ 647	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-647}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{14}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{14}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{275}a^{20}-\frac{3}{55}a^{19}+\frac{1}{25}a^{18}+\frac{6}{275}a^{17}+\frac{2}{25}a^{16}-\frac{4}{55}a^{15}+\frac{54}{275}a^{14}+\frac{1}{55}a^{13}-\frac{12}{25}a^{12}+\frac{42}{275}a^{11}+\frac{47}{275}a^{10}+\frac{42}{275}a^{9}-\frac{91}{275}a^{8}+\frac{73}{275}a^{7}-\frac{97}{275}a^{6}-\frac{73}{275}a^{5}+\frac{23}{275}a^{4}+\frac{26}{275}a^{3}+\frac{68}{275}a^{2}+\frac{39}{275}a+\frac{119}{275}$, $\frac{1}{193325}a^{21}+\frac{6}{17575}a^{20}-\frac{19079}{193325}a^{19}-\frac{8948}{193325}a^{18}-\frac{15937}{193325}a^{17}+\frac{9352}{193325}a^{16}+\frac{14164}{193325}a^{15}+\frac{93754}{193325}a^{14}-\frac{4952}{193325}a^{13}-\frac{5474}{38665}a^{12}+\frac{60099}{193325}a^{11}+\frac{52}{5225}a^{10}-\frac{8509}{17575}a^{9}+\frac{55512}{193325}a^{8}-\frac{17779}{193325}a^{7}-\frac{1212}{38665}a^{6}-\frac{12332}{38665}a^{5}+\frac{71629}{193325}a^{4}+\frac{47549}{193325}a^{3}+\frac{3233}{10175}a^{2}-\frac{187}{17575}a+\frac{15414}{193325}$, $\frac{1}{16889065325}a^{22}+\frac{6236}{16889065325}a^{21}+\frac{15050612}{16889065325}a^{20}+\frac{25039191}{456461225}a^{19}+\frac{1541886859}{16889065325}a^{18}+\frac{233319718}{16889065325}a^{17}+\frac{7588288}{456461225}a^{16}+\frac{934312404}{16889065325}a^{15}-\frac{4959075178}{16889065325}a^{14}-\frac{78682011}{3377813065}a^{13}+\frac{536483197}{16889065325}a^{12}+\frac{689341351}{16889065325}a^{11}+\frac{4786760163}{16889065325}a^{10}+\frac{148378891}{888898175}a^{9}+\frac{644631076}{3377813065}a^{8}+\frac{8099028858}{16889065325}a^{7}+\frac{5025800893}{16889065325}a^{6}+\frac{5105693096}{16889065325}a^{5}+\frac{1067482887}{16889065325}a^{4}-\frac{2095142937}{16889065325}a^{3}-\frac{4610451389}{16889065325}a^{2}-\frac{6103940692}{16889065325}a-\frac{3711736931}{16889065325}$ 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, 1/5*a^15 + 2/5*a^13 - 1/5*a^12 - 1/5*a^11 + 2/5*a^10 - 2/5*a^8 + 2/5*a^7 - 2/5*a^6 + 2/5*a^5 + 1/5*a^4 + 1/5*a^3 - 1/5*a^2 - 2/5*a - 2/5, 1/5*a^16 + 2/5*a^14 - 1/5*a^13 - 1/5*a^12 + 2/5*a^11 - 2/5*a^9 + 2/5*a^8 - 2/5*a^7 + 2/5*a^6 + 1/5*a^5 + 1/5*a^4 - 1/5*a^3 - 2/5*a^2 - 2/5*a, 1/5*a^17 - 1/5*a^14 - 1/5*a^12 + 2/5*a^11 - 1/5*a^10 + 2/5*a^9 + 2/5*a^8 - 2/5*a^7 + 2/5*a^5 + 2/5*a^4 + 1/5*a^3 - 1/5*a - 1/5, 1/5*a^18 + 1/5*a^13 + 1/5*a^12 - 2/5*a^11 - 1/5*a^10 + 2/5*a^9 + 1/5*a^8 + 2/5*a^7 - 1/5*a^5 + 2/5*a^4 + 1/5*a^3 - 2/5*a^2 + 2/5*a - 2/5, 1/5*a^19 + 1/5*a^14 + 1/5*a^13 - 2/5*a^12 - 1/5*a^11 + 2/5*a^10 + 1/5*a^9 + 2/5*a^8 - 1/5*a^6 + 2/5*a^5 + 1/5*a^4 - 2/5*a^3 + 2/5*a^2 - 2/5*a, 1/275*a^20 - 3/55*a^19 + 1/25*a^18 + 6/275*a^17 + 2/25*a^16 - 4/55*a^15 + 54/275*a^14 + 1/55*a^13 - 12/25*a^12 + 42/275*a^11 + 47/275*a^10 + 42/275*a^9 - 91/275*a^8 + 73/275*a^7 - 97/275*a^6 - 73/275*a^5 + 23/275*a^4 + 26/275*a^3 + 68/275*a^2 + 39/275*a + 119/275, 1/193325*a^21 + 6/17575*a^20 - 19079/193325*a^19 - 8948/193325*a^18 - 15937/193325*a^17 + 9352/193325*a^16 + 14164/193325*a^15 + 93754/193325*a^14 - 4952/193325*a^13 - 5474/38665*a^12 + 60099/193325*a^11 + 52/5225*a^10 - 8509/17575*a^9 + 55512/193325*a^8 - 17779/193325*a^7 - 1212/38665*a^6 - 12332/38665*a^5 + 71629/193325*a^4 + 47549/193325*a^3 + 3233/10175*a^2 - 187/17575*a + 15414/193325, 1/16889065325*a^22 + 6236/16889065325*a^21 + 15050612/16889065325*a^20 + 25039191/456461225*a^19 + 1541886859/16889065325*a^18 + 233319718/16889065325*a^17 + 7588288/456461225*a^16 + 934312404/16889065325*a^15 - 4959075178/16889065325*a^14 - 78682011/3377813065*a^13 + 536483197/16889065325*a^12 + 689341351/16889065325*a^11 + 4786760163/16889065325*a^10 + 148378891/888898175*a^9 + 644631076/3377813065*a^8 + 8099028858/16889065325*a^7 + 5025800893/16889065325*a^6 + 5105693096/16889065325*a^5 + 1067482887/16889065325*a^4 - 2095142937/16889065325*a^3 - 4610451389/16889065325*a^2 - 6103940692/16889065325*a - 3711736931/16889065325

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

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:	$\frac{812678413}{16889065325}a^{22}-\frac{52028243}{456461225}a^{21}-\frac{3228708099}{16889065325}a^{20}+\frac{12328573847}{16889065325}a^{19}-\frac{13615730822}{16889065325}a^{18}+\frac{12920257576}{16889065325}a^{17}+\frac{4790191933}{16889065325}a^{16}-\frac{74609096329}{16889065325}a^{15}+\frac{168858186111}{16889065325}a^{14}-\frac{288887910317}{16889065325}a^{13}+\frac{434708175028}{16889065325}a^{12}-\frac{107760037102}{3377813065}a^{11}+\frac{597451945736}{16889065325}a^{10}-\frac{571916535374}{16889065325}a^{9}+\frac{374789573088}{16889065325}a^{8}-\frac{185246864343}{16889065325}a^{7}+\frac{39627989746}{16889065325}a^{6}+\frac{92061047556}{16889065325}a^{5}-\frac{136216139643}{16889065325}a^{4}+\frac{95378875127}{16889065325}a^{3}-\frac{65997598053}{16889065325}a^{2}-\frac{12641836642}{16889065325}a+\frac{14428923637}{16889065325}$, $\frac{9390537}{456461225}a^{22}-\frac{663331444}{16889065325}a^{21}-\frac{1197685008}{16889065325}a^{20}+\frac{826078404}{3377813065}a^{19}-\frac{5414037143}{16889065325}a^{18}+\frac{1494044914}{3377813065}a^{17}-\frac{145386388}{1535369575}a^{16}-\frac{23146907106}{16889065325}a^{15}+\frac{61341834594}{16889065325}a^{14}-\frac{122371478664}{16889065325}a^{13}+\frac{10883524126}{888898175}a^{12}-\frac{290172111149}{16889065325}a^{11}+\frac{10085389522}{456461225}a^{10}-\frac{421781506568}{16889065325}a^{9}+\frac{21275231518}{888898175}a^{8}-\frac{14190122588}{675562613}a^{7}+\frac{284006985763}{16889065325}a^{6}-\frac{16607085982}{1535369575}a^{5}+\frac{95443715027}{16889065325}a^{4}-\frac{48946684448}{16889065325}a^{3}+\frac{20145952509}{16889065325}a^{2}-\frac{9705555284}{16889065325}a+\frac{8607536032}{16889065325}$, $\frac{1196627247}{16889065325}a^{22}-\frac{1590721923}{16889065325}a^{21}-\frac{11140604}{41496475}a^{20}+\frac{1157554774}{1535369575}a^{19}-\frac{15040889964}{16889065325}a^{18}+\frac{1309223204}{1535369575}a^{17}+\frac{11705125118}{16889065325}a^{16}-\frac{4350921273}{888898175}a^{15}+\frac{189160213031}{16889065325}a^{14}-\frac{5927058559}{307073915}a^{13}+\frac{483911484528}{16889065325}a^{12}-\frac{616939708742}{16889065325}a^{11}+\frac{713010641427}{16889065325}a^{10}-\frac{692570128566}{16889065325}a^{9}+\frac{546926257342}{16889065325}a^{8}-\frac{72920050649}{3377813065}a^{7}+\frac{4081476159}{307073915}a^{6}-\frac{69044614522}{16889065325}a^{5}-\frac{8892378672}{16889065325}a^{4}+\frac{21759548754}{16889065325}a^{3}+\frac{6392145431}{16889065325}a^{2}-\frac{150475612}{1535369575}a+\frac{3629332609}{3377813065}$, $\frac{369638233}{16889065325}a^{22}-\frac{447893803}{16889065325}a^{21}-\frac{446581077}{16889065325}a^{20}+\frac{69202059}{307073915}a^{19}-\frac{8433612047}{16889065325}a^{18}+\frac{723717199}{1535369575}a^{17}-\frac{987586103}{16889065325}a^{16}-\frac{17958153847}{16889065325}a^{15}+\frac{72321493939}{16889065325}a^{14}-\frac{12727132043}{1535369575}a^{13}+\frac{2372997376}{177779635}a^{12}-\frac{65259370038}{3377813065}a^{11}+\frac{402841722131}{16889065325}a^{10}-\frac{437523977533}{16889065325}a^{9}+\frac{4448114263}{177779635}a^{8}-\frac{317031831023}{16889065325}a^{7}+\frac{225081459688}{16889065325}a^{6}-\frac{128714509201}{16889065325}a^{5}+\frac{34464632701}{16889065325}a^{4}+\frac{21953236653}{16889065325}a^{3}-\frac{1366407987}{3377813065}a^{2}+\frac{9315775608}{16889065325}a+\frac{1672875318}{3377813065}$, $\frac{207435893}{16889065325}a^{22}-\frac{19519860}{675562613}a^{21}-\frac{18018414}{675562613}a^{20}+\frac{2561749863}{16889065325}a^{19}-\frac{4906003247}{16889065325}a^{18}+\frac{7094794792}{16889065325}a^{17}-\frac{3643309639}{16889065325}a^{16}-\frac{2406688436}{3377813065}a^{15}+\frac{43937413262}{16889065325}a^{14}-\frac{100058423209}{16889065325}a^{13}+\frac{170153332097}{16889065325}a^{12}-\frac{50114388593}{3377813065}a^{11}+\frac{336326507916}{16889065325}a^{10}-\frac{35276770772}{1535369575}a^{9}+\frac{388019589237}{16889065325}a^{8}-\frac{356853459913}{16889065325}a^{7}+\frac{54136960971}{3377813065}a^{6}-\frac{17168189473}{1535369575}a^{5}+\frac{20003715994}{3377813065}a^{4}-\frac{26966076131}{16889065325}a^{3}+\frac{3037797028}{16889065325}a^{2}-\frac{5742606787}{16889065325}a-\frac{24023137647}{16889065325}$, $\frac{234573138}{3377813065}a^{22}-\frac{168101005}{675562613}a^{21}-\frac{3620846631}{16889065325}a^{20}+\frac{933318659}{675562613}a^{19}-\frac{30761491866}{16889065325}a^{18}+\frac{32219921809}{16889065325}a^{17}-\frac{7582812312}{16889065325}a^{16}-\frac{5058449178}{675562613}a^{15}+\frac{326995898751}{16889065325}a^{14}-\frac{23895550462}{675562613}a^{13}+\frac{944293810417}{16889065325}a^{12}-\frac{1233730788122}{16889065325}a^{11}+\frac{1433025610253}{16889065325}a^{10}-\frac{1498183900757}{16889065325}a^{9}+\frac{1157894608171}{16889065325}a^{8}-\frac{69418363918}{1535369575}a^{7}+\frac{412820509007}{16889065325}a^{6}-\frac{71627550227}{16889065325}a^{5}-\frac{12185731978}{1535369575}a^{4}+\frac{7519060944}{1535369575}a^{3}-\frac{121152684858}{16889065325}a^{2}-\frac{35779991239}{16889065325}a+\frac{13311409736}{16889065325}$, $\frac{550483487}{16889065325}a^{22}-\frac{148216973}{3377813065}a^{21}-\frac{2221153391}{16889065325}a^{20}+\frac{6358804132}{16889065325}a^{19}-\frac{6526310059}{16889065325}a^{18}+\frac{128621976}{456461225}a^{17}+\frac{8876523357}{16889065325}a^{16}-\frac{8803971624}{3377813065}a^{15}+\frac{90513711669}{16889065325}a^{14}-\frac{142196903141}{16889065325}a^{13}+\frac{38614140589}{3377813065}a^{12}-\frac{206664504487}{16889065325}a^{11}+\frac{195828095432}{16889065325}a^{10}-\frac{1271561378}{177779635}a^{9}-\frac{13799549336}{16889065325}a^{8}+\frac{27285865874}{3377813065}a^{7}-\frac{193171124758}{16889065325}a^{6}+\frac{228384578706}{16889065325}a^{5}-\frac{187895782933}{16889065325}a^{4}+\frac{4998830106}{675562613}a^{3}-\frac{39627791836}{16889065325}a^{2}-\frac{17963692497}{16889065325}a+\frac{20062533473}{16889065325}$, $\frac{532290973}{16889065325}a^{22}-\frac{516255218}{16889065325}a^{21}-\frac{2143291206}{16889065325}a^{20}+\frac{27990431}{91292245}a^{19}-\frac{5224213801}{16889065325}a^{18}+\frac{685984752}{3377813065}a^{17}+\frac{12076558}{24024275}a^{16}-\frac{3156303462}{1535369575}a^{15}+\frac{72318286418}{16889065325}a^{14}-\frac{114854421483}{16889065325}a^{13}+\frac{160550279063}{16889065325}a^{12}-\frac{195673287393}{16889065325}a^{11}+\frac{220436450213}{16889065325}a^{10}-\frac{199855214126}{16889065325}a^{9}+\frac{151298552249}{16889065325}a^{8}-\frac{20250524731}{3377813065}a^{7}+\frac{83891146796}{16889065325}a^{6}-\frac{64119126444}{16889065325}a^{5}+\frac{4568664274}{1535369575}a^{4}-\frac{32950079896}{16889065325}a^{3}+\frac{36005769313}{16889065325}a^{2}-\frac{24630951098}{16889065325}a+\frac{385111679}{1535369575}$, $\frac{89142651}{3377813065}a^{22}-\frac{2318319606}{16889065325}a^{21}-\frac{801230651}{16889065325}a^{20}+\frac{1165553954}{1535369575}a^{19}-\frac{17534375477}{16889065325}a^{18}+\frac{1301187567}{1535369575}a^{17}-\frac{4363437902}{16889065325}a^{16}-\frac{61691680929}{16889065325}a^{15}+\frac{169481486961}{16889065325}a^{14}-\frac{26996473348}{1535369575}a^{13}+\frac{93647457133}{3377813065}a^{12}-\frac{611749331399}{16889065325}a^{11}+\frac{702055487461}{16889065325}a^{10}-\frac{740695491746}{16889065325}a^{9}+\frac{570103624878}{16889065325}a^{8}-\frac{342227151461}{16889065325}a^{7}+\frac{39833145933}{3377813065}a^{6}-\frac{12128341778}{3377813065}a^{5}-\frac{41394807639}{16889065325}a^{4}+\frac{53145833441}{16889065325}a^{3}-\frac{59940211237}{16889065325}a^{2}+\frac{8881961412}{16889065325}a+\frac{14940317321}{16889065325}$, $\frac{762189006}{16889065325}a^{22}-\frac{437515987}{3377813065}a^{21}-\frac{63932214}{675562613}a^{20}+\frac{12404603601}{16889065325}a^{19}-\frac{564359427}{456461225}a^{18}+\frac{24207117064}{16889065325}a^{17}-\frac{6174497658}{16889065325}a^{16}-\frac{249176756}{61414783}a^{15}+\frac{199202350234}{16889065325}a^{14}-\frac{10426356099}{456461225}a^{13}+\frac{32402727536}{888898175}a^{12}-\frac{165388342727}{3377813065}a^{11}+\frac{993856695102}{16889065325}a^{10}-\frac{1049412987764}{16889065325}a^{9}+\frac{47349453281}{888898175}a^{8}-\frac{643275390506}{16889065325}a^{7}+\frac{74991964987}{3377813065}a^{6}-\frac{146215103966}{16889065325}a^{5}+\frac{4765085802}{3377813065}a^{4}+\frac{17745033338}{16889065325}a^{3}-\frac{31396188339}{16889065325}a^{2}-\frac{7422477559}{16889065325}a+\frac{7953930686}{16889065325}$, $\frac{2108968274}{16889065325}a^{22}-\frac{6874150726}{16889065325}a^{21}-\frac{697485867}{3377813065}a^{20}+\frac{39017827393}{16889065325}a^{19}-\frac{63885101187}{16889065325}a^{18}+\frac{68045806479}{16889065325}a^{17}-\frac{28995892252}{16889065325}a^{16}-\frac{188623022729}{16889065325}a^{15}+\frac{602436179921}{16889065325}a^{14}-\frac{230512267289}{3377813065}a^{13}+\frac{1880537761914}{16889065325}a^{12}-\frac{136873520653}{888898175}a^{11}+\frac{3140034680806}{16889065325}a^{10}-\frac{18396595881}{91292245}a^{9}+\frac{3009596243818}{16889065325}a^{8}-\frac{2178868485802}{16889065325}a^{7}+\frac{1444307788798}{16889065325}a^{6}-\frac{654995315152}{16889065325}a^{5}+\frac{34637598269}{16889065325}a^{4}+\frac{124584830654}{16889065325}a^{3}-\frac{5339619055}{675562613}a^{2}+\frac{7039542669}{3377813065}a+\frac{74605410799}{16889065325}$ 812678413/16889065325a^22 - 52028243/456461225a^21 - 3228708099/16889065325a^20 + 12328573847/16889065325a^19 - 13615730822/16889065325a^18 + 12920257576/16889065325a^17 + 4790191933/16889065325a^16 - 74609096329/16889065325a^15 + 168858186111/16889065325a^14 - 288887910317/16889065325a^13 + 434708175028/16889065325a^12 - 107760037102/3377813065a^11 + 597451945736/16889065325a^10 - 571916535374/16889065325a^9 + 374789573088/16889065325a^8 - 185246864343/16889065325a^7 + 39627989746/16889065325a^6 + 92061047556/16889065325a^5 - 136216139643/16889065325a^4 + 95378875127/16889065325a^3 - 65997598053/16889065325a^2 - 12641836642/16889065325a + 14428923637/16889065325, 9390537/456461225a^22 - 663331444/16889065325a^21 - 1197685008/16889065325a^20 + 826078404/3377813065a^19 - 5414037143/16889065325a^18 + 1494044914/3377813065a^17 - 145386388/1535369575a^16 - 23146907106/16889065325a^15 + 61341834594/16889065325a^14 - 122371478664/16889065325a^13 + 10883524126/888898175a^12 - 290172111149/16889065325a^11 + 10085389522/456461225a^10 - 421781506568/16889065325a^9 + 21275231518/888898175a^8 - 14190122588/675562613a^7 + 284006985763/16889065325a^6 - 16607085982/1535369575a^5 + 95443715027/16889065325a^4 - 48946684448/16889065325a^3 + 20145952509/16889065325a^2 - 9705555284/16889065325a + 8607536032/16889065325, 1196627247/16889065325a^22 - 1590721923/16889065325a^21 - 11140604/41496475a^20 + 1157554774/1535369575a^19 - 15040889964/16889065325a^18 + 1309223204/1535369575a^17 + 11705125118/16889065325a^16 - 4350921273/888898175a^15 + 189160213031/16889065325a^14 - 5927058559/307073915a^13 + 483911484528/16889065325a^12 - 616939708742/16889065325a^11 + 713010641427/16889065325a^10 - 692570128566/16889065325a^9 + 546926257342/16889065325a^8 - 72920050649/3377813065a^7 + 4081476159/307073915a^6 - 69044614522/16889065325a^5 - 8892378672/16889065325a^4 + 21759548754/16889065325a^3 + 6392145431/16889065325a^2 - 150475612/1535369575a + 3629332609/3377813065, 369638233/16889065325a^22 - 447893803/16889065325a^21 - 446581077/16889065325a^20 + 69202059/307073915a^19 - 8433612047/16889065325a^18 + 723717199/1535369575a^17 - 987586103/16889065325a^16 - 17958153847/16889065325a^15 + 72321493939/16889065325a^14 - 12727132043/1535369575a^13 + 2372997376/177779635a^12 - 65259370038/3377813065a^11 + 402841722131/16889065325a^10 - 437523977533/16889065325a^9 + 4448114263/177779635a^8 - 317031831023/16889065325a^7 + 225081459688/16889065325a^6 - 128714509201/16889065325a^5 + 34464632701/16889065325a^4 + 21953236653/16889065325a^3 - 1366407987/3377813065a^2 + 9315775608/16889065325a + 1672875318/3377813065, 207435893/16889065325a^22 - 19519860/675562613a^21 - 18018414/675562613a^20 + 2561749863/16889065325a^19 - 4906003247/16889065325a^18 + 7094794792/16889065325a^17 - 3643309639/16889065325a^16 - 2406688436/3377813065a^15 + 43937413262/16889065325a^14 - 100058423209/16889065325a^13 + 170153332097/16889065325a^12 - 50114388593/3377813065a^11 + 336326507916/16889065325a^10 - 35276770772/1535369575a^9 + 388019589237/16889065325a^8 - 356853459913/16889065325a^7 + 54136960971/3377813065a^6 - 17168189473/1535369575a^5 + 20003715994/3377813065a^4 - 26966076131/16889065325a^3 + 3037797028/16889065325a^2 - 5742606787/16889065325a - 24023137647/16889065325, 234573138/3377813065a^22 - 168101005/675562613a^21 - 3620846631/16889065325a^20 + 933318659/675562613a^19 - 30761491866/16889065325a^18 + 32219921809/16889065325a^17 - 7582812312/16889065325a^16 - 5058449178/675562613a^15 + 326995898751/16889065325a^14 - 23895550462/675562613a^13 + 944293810417/16889065325a^12 - 1233730788122/16889065325a^11 + 1433025610253/16889065325a^10 - 1498183900757/16889065325a^9 + 1157894608171/16889065325a^8 - 69418363918/1535369575a^7 + 412820509007/16889065325a^6 - 71627550227/16889065325a^5 - 12185731978/1535369575a^4 + 7519060944/1535369575a^3 - 121152684858/16889065325a^2 - 35779991239/16889065325a + 13311409736/16889065325, 550483487/16889065325a^22 - 148216973/3377813065a^21 - 2221153391/16889065325a^20 + 6358804132/16889065325a^19 - 6526310059/16889065325a^18 + 128621976/456461225a^17 + 8876523357/16889065325a^16 - 8803971624/3377813065a^15 + 90513711669/16889065325a^14 - 142196903141/16889065325a^13 + 38614140589/3377813065a^12 - 206664504487/16889065325a^11 + 195828095432/16889065325a^10 - 1271561378/177779635a^9 - 13799549336/16889065325a^8 + 27285865874/3377813065a^7 - 193171124758/16889065325a^6 + 228384578706/16889065325a^5 - 187895782933/16889065325a^4 + 4998830106/675562613a^3 - 39627791836/16889065325a^2 - 17963692497/16889065325a + 20062533473/16889065325, 532290973/16889065325a^22 - 516255218/16889065325a^21 - 2143291206/16889065325a^20 + 27990431/91292245a^19 - 5224213801/16889065325a^18 + 685984752/3377813065a^17 + 12076558/24024275a^16 - 3156303462/1535369575a^15 + 72318286418/16889065325a^14 - 114854421483/16889065325a^13 + 160550279063/16889065325a^12 - 195673287393/16889065325a^11 + 220436450213/16889065325a^10 - 199855214126/16889065325a^9 + 151298552249/16889065325a^8 - 20250524731/3377813065a^7 + 83891146796/16889065325a^6 - 64119126444/16889065325a^5 + 4568664274/1535369575a^4 - 32950079896/16889065325a^3 + 36005769313/16889065325a^2 - 24630951098/16889065325a + 385111679/1535369575, 89142651/3377813065a^22 - 2318319606/16889065325a^21 - 801230651/16889065325a^20 + 1165553954/1535369575a^19 - 17534375477/16889065325a^18 + 1301187567/1535369575a^17 - 4363437902/16889065325a^16 - 61691680929/16889065325a^15 + 169481486961/16889065325a^14 - 26996473348/1535369575a^13 + 93647457133/3377813065a^12 - 611749331399/16889065325a^11 + 702055487461/16889065325a^10 - 740695491746/16889065325a^9 + 570103624878/16889065325a^8 - 342227151461/16889065325a^7 + 39833145933/3377813065a^6 - 12128341778/3377813065a^5 - 41394807639/16889065325a^4 + 53145833441/16889065325a^3 - 59940211237/16889065325a^2 + 8881961412/16889065325a + 14940317321/16889065325, 762189006/16889065325a^22 - 437515987/3377813065a^21 - 63932214/675562613a^20 + 12404603601/16889065325a^19 - 564359427/456461225a^18 + 24207117064/16889065325a^17 - 6174497658/16889065325a^16 - 249176756/61414783a^15 + 199202350234/16889065325a^14 - 10426356099/456461225a^13 + 32402727536/888898175a^12 - 165388342727/3377813065a^11 + 993856695102/16889065325a^10 - 1049412987764/16889065325a^9 + 47349453281/888898175a^8 - 643275390506/16889065325a^7 + 74991964987/3377813065a^6 - 146215103966/16889065325a^5 + 4765085802/3377813065a^4 + 17745033338/16889065325a^3 - 31396188339/16889065325a^2 - 7422477559/16889065325a + 7953930686/16889065325, 2108968274/16889065325a^22 - 6874150726/16889065325a^21 - 697485867/3377813065a^20 + 39017827393/16889065325a^19 - 63885101187/16889065325a^18 + 68045806479/16889065325a^17 - 28995892252/16889065325a^16 - 188623022729/16889065325a^15 + 602436179921/16889065325a^14 - 230512267289/3377813065a^13 + 1880537761914/16889065325a^12 - 136873520653/888898175a^11 + 3140034680806/16889065325a^10 - 18396595881/91292245a^9 + 3009596243818/16889065325a^8 - 2178868485802/16889065325a^7 + 1444307788798/16889065325a^6 - 654995315152/16889065325a^5 + 34637598269/16889065325a^4 + 124584830654/16889065325a^3 - 5339619055/675562613a^2 + 7039542669/3377813065a + 74605410799/16889065325 (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:	$ 1561186.27544 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 1561186.27544 \cdot 1}{2\cdot\sqrt{8316624386164136054632388935703}}\cr\approx \mathstrut & 0.326181805972 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 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^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 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^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 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^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 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

$D_{23}$ (as 23T2):

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 46

The 13 conjugacy class representatives for $D_{23}$

Character table for $D_{23}$

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

Galois closure:	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	$23$	$23$	${\href{/padicField/5.2.0.1}{2} }^{11}{,}\,{\href{/padicField/5.1.0.1}{1} }$	$23$	${\href{/padicField/11.2.0.1}{2} }^{11}{,}\,{\href{/padicField/11.1.0.1}{1} }$	$23$	$23$	${\href{/padicField/19.2.0.1}{2} }^{11}{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.2.0.1}{2} }^{11}{,}\,{\href{/padicField/23.1.0.1}{1} }$	$23$	$23$	${\href{/padicField/37.2.0.1}{2} }^{11}{,}\,{\href{/padicField/37.1.0.1}{1} }$	$23$	$23$	${\href{/padicField/47.2.0.1}{2} }^{11}{,}\,{\href{/padicField/47.1.0.1}{1} }$	$23$	${\href{/padicField/59.2.0.1}{2} }^{11}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$647$ 647	$\Q_{647}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Overview	Random
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
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Abelian varieties over $\F_{q}$

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