Properties

Label 23.1.582...903.1
Degree $23$
Signature $[1, 11]$
Discriminant $-5.823\times 10^{34}$
Root discriminant \(32.47\)
Ramified prime $1447$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{23}$ (as 23T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*x + 1)
 
gp: K = bnfinit(y^23 - 2*y^22 + 16*y^21 - 34*y^20 + 68*y^19 - 63*y^18 + 17*y^17 + 43*y^16 - 139*y^15 + 165*y^14 + 25*y^13 - 228*y^12 + 265*y^11 - 270*y^10 + 75*y^9 + 246*y^8 - 130*y^7 - 161*y^6 + 240*y^5 - 393*y^4 + 569*y^3 - 385*y^2 + 99*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 2*x^22 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*x + 1)
 

\( x^{23} - 2 x^{22} + 16 x^{21} - 34 x^{20} + 68 x^{19} - 63 x^{18} + 17 x^{17} + 43 x^{16} - 139 x^{15} + 165 x^{14} + 25 x^{13} - 228 x^{12} + 265 x^{11} - 270 x^{10} + 75 x^{9} + 246 x^{8} + \cdots + 1 \) Copy content Toggle raw display

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:   \(-58230945563524325903440619509366903\) \(\medspace = -\,1447^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(32.47\)
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:  $1447^{1/2}\approx 38.03945320322047$
Ramified primes:   \(1447\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1447}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{13}+\frac{4}{27}a^{12}-\frac{1}{9}a^{11}-\frac{4}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{9}a^{8}+\frac{10}{27}a^{6}-\frac{7}{27}a^{5}-\frac{4}{9}a^{4}+\frac{4}{27}a^{3}+\frac{5}{27}a^{2}+\frac{1}{3}a-\frac{2}{27}$, $\frac{1}{27}a^{16}-\frac{1}{27}a^{14}+\frac{4}{27}a^{13}-\frac{1}{9}a^{12}-\frac{4}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{9}a^{9}+\frac{1}{27}a^{7}+\frac{11}{27}a^{6}+\frac{2}{9}a^{5}-\frac{5}{27}a^{4}-\frac{4}{27}a^{3}-\frac{11}{27}a-\frac{1}{3}$, $\frac{1}{27}a^{17}+\frac{1}{27}a^{14}-\frac{1}{27}a^{13}+\frac{4}{27}a^{11}+\frac{2}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{4}{27}a^{7}+\frac{4}{27}a^{6}-\frac{1}{9}a^{5}-\frac{4}{27}a^{4}+\frac{10}{27}a^{3}+\frac{1}{9}a^{2}+\frac{4}{9}a+\frac{13}{27}$, $\frac{1}{135}a^{18}-\frac{2}{135}a^{17}+\frac{1}{135}a^{15}-\frac{19}{135}a^{13}+\frac{13}{135}a^{12}+\frac{1}{9}a^{11}+\frac{4}{45}a^{10}+\frac{17}{135}a^{9}-\frac{4}{45}a^{8}+\frac{1}{15}a^{7}+\frac{64}{135}a^{6}-\frac{7}{135}a^{5}-\frac{7}{45}a^{4}+\frac{8}{27}a^{3}-\frac{13}{45}a^{2}+\frac{49}{135}a+\frac{4}{135}$, $\frac{1}{135}a^{19}+\frac{1}{135}a^{17}+\frac{1}{135}a^{16}+\frac{2}{135}a^{15}+\frac{1}{135}a^{14}-\frac{4}{135}a^{12}-\frac{13}{135}a^{11}-\frac{1}{15}a^{10}-\frac{2}{15}a^{9}+\frac{1}{27}a^{8}+\frac{2}{135}a^{7}+\frac{22}{45}a^{6}-\frac{1}{27}a^{5}-\frac{37}{135}a^{4}-\frac{29}{135}a^{3}-\frac{14}{135}a^{2}+\frac{4}{45}a-\frac{2}{135}$, $\frac{1}{405}a^{20}-\frac{1}{405}a^{19}-\frac{1}{135}a^{17}+\frac{1}{405}a^{16}-\frac{2}{405}a^{15}-\frac{7}{135}a^{14}-\frac{11}{81}a^{13}-\frac{67}{405}a^{12}-\frac{1}{405}a^{11}-\frac{16}{405}a^{10}-\frac{44}{405}a^{9}+\frac{34}{405}a^{8}+\frac{1}{9}a^{7}+\frac{20}{81}a^{6}+\frac{7}{81}a^{5}+\frac{199}{405}a^{4}+\frac{1}{81}a^{2}+\frac{1}{15}a+\frac{8}{405}$, $\frac{1}{2025}a^{21}+\frac{2}{2025}a^{19}-\frac{1}{405}a^{17}+\frac{17}{2025}a^{16}-\frac{14}{2025}a^{15}+\frac{92}{2025}a^{14}+\frac{241}{2025}a^{13}-\frac{86}{2025}a^{12}+\frac{244}{2025}a^{11}-\frac{8}{75}a^{10}+\frac{32}{2025}a^{9}+\frac{103}{2025}a^{8}+\frac{13}{2025}a^{7}-\frac{32}{135}a^{6}+\frac{62}{225}a^{5}-\frac{46}{405}a^{4}-\frac{112}{2025}a^{3}-\frac{937}{2025}a^{2}-\frac{127}{2025}a-\frac{346}{2025}$, $\frac{1}{104470934408325}a^{22}-\frac{914858467}{6964728960555}a^{21}-\frac{32762334668}{104470934408325}a^{20}-\frac{56589028498}{20894186881665}a^{19}-\frac{32608635196}{20894186881665}a^{18}+\frac{298230323552}{104470934408325}a^{17}-\frac{175588356248}{34823644802775}a^{16}-\frac{965948654048}{104470934408325}a^{15}+\frac{50294744476}{104470934408325}a^{14}+\frac{8343057920609}{104470934408325}a^{13}-\frac{15682282111}{104470934408325}a^{12}+\frac{1391427574309}{104470934408325}a^{11}-\frac{401488707002}{11607881600925}a^{10}-\frac{737935559353}{11607881600925}a^{9}+\frac{5414916967156}{34823644802775}a^{8}-\frac{150434843501}{2321576320185}a^{7}-\frac{13838276284492}{104470934408325}a^{6}+\frac{3788513561851}{20894186881665}a^{5}+\frac{38665881240733}{104470934408325}a^{4}-\frac{1968498721702}{104470934408325}a^{3}+\frac{7920828501611}{34823644802775}a^{2}-\frac{8471973936001}{104470934408325}a+\frac{6111746943359}{20894186881665}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$, $5$

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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{633691164013}{6964728960555}a^{22}-\frac{187652154623}{1392945792111}a^{21}+\frac{9635269559323}{6964728960555}a^{20}-\frac{16538481015637}{6964728960555}a^{19}+\frac{34234475496106}{6964728960555}a^{18}-\frac{21955353114377}{6964728960555}a^{17}-\frac{346603230763}{1392945792111}a^{16}+\frac{9085236501371}{2321576320185}a^{15}-\frac{76416663464029}{6964728960555}a^{14}+\frac{67082022822988}{6964728960555}a^{13}+\frac{50665909952251}{6964728960555}a^{12}-\frac{2658245781557}{154771754679}a^{11}+\frac{36654276033472}{2321576320185}a^{10}-\frac{121507848745502}{6964728960555}a^{9}-\frac{2590476189242}{1392945792111}a^{8}+\frac{155016079546184}{6964728960555}a^{7}-\frac{7615859221342}{6964728960555}a^{6}-\frac{95505354036952}{6964728960555}a^{5}+\frac{10156132525939}{773858773395}a^{4}-\frac{2489469923361}{85984308155}a^{3}+\frac{49567706279429}{1392945792111}a^{2}-\frac{6637329575035}{464315264037}a+\frac{1048384799716}{1392945792111}$, $\frac{7947616666166}{104470934408325}a^{22}-\frac{1098369443522}{11607881600925}a^{21}+\frac{116965795699367}{104470934408325}a^{20}-\frac{179953153150906}{104470934408325}a^{19}+\frac{73022582210377}{20894186881665}a^{18}-\frac{185442855163223}{104470934408325}a^{17}-\frac{7393178857577}{6964728960555}a^{16}+\frac{264528811181189}{104470934408325}a^{15}-\frac{173738952548759}{20894186881665}a^{14}+\frac{592329468504571}{104470934408325}a^{13}+\frac{918422801992882}{104470934408325}a^{12}-\frac{11\!\cdots\!03}{104470934408325}a^{11}+\frac{12722592151091}{1392945792111}a^{10}-\frac{133905245848672}{11607881600925}a^{9}-\frac{182552833288742}{34823644802775}a^{8}+\frac{601865817206227}{34823644802775}a^{7}+\frac{626680085384203}{104470934408325}a^{6}-\frac{10\!\cdots\!69}{104470934408325}a^{5}+\frac{715103529743048}{104470934408325}a^{4}-\frac{24\!\cdots\!16}{104470934408325}a^{3}+\frac{830072057011853}{34823644802775}a^{2}-\frac{109653275938822}{20894186881665}a-\frac{84907657231177}{104470934408325}$, $\frac{151490495887}{11607881600925}a^{22}-\frac{175567805237}{3869293866975}a^{21}+\frac{7140625923907}{34823644802775}a^{20}-\frac{24734157480661}{34823644802775}a^{19}+\frac{2142187953326}{2321576320185}a^{18}-\frac{15623019864301}{11607881600925}a^{17}-\frac{2865283207498}{6964728960555}a^{16}+\frac{21197177506204}{34823644802775}a^{15}-\frac{4723962955517}{2321576320185}a^{14}+\frac{104325825554696}{34823644802775}a^{13}+\frac{40135292076227}{34823644802775}a^{12}-\frac{176995501754758}{34823644802775}a^{11}+\frac{3445277207486}{1392945792111}a^{10}-\frac{80704023526943}{34823644802775}a^{9}+\frac{83809076477794}{34823644802775}a^{8}+\frac{56440365815542}{11607881600925}a^{7}-\frac{28655066753927}{34823644802775}a^{6}-\frac{269169983988184}{34823644802775}a^{5}+\frac{24952041265198}{34823644802775}a^{4}-\frac{24669994987532}{11607881600925}a^{3}+\frac{337034824310384}{34823644802775}a^{2}-\frac{16363591195594}{2321576320185}a+\frac{33019104036023}{34823644802775}$, $\frac{2617988812}{3869293866975}a^{22}+\frac{5773002847}{3869293866975}a^{21}+\frac{687542004491}{34823644802775}a^{20}+\frac{454294451626}{34823644802775}a^{19}+\frac{126827257786}{773858773395}a^{18}-\frac{22001254649}{429921540775}a^{17}+\frac{16056504855209}{34823644802775}a^{16}+\frac{3634529526209}{34823644802775}a^{15}+\frac{695411782433}{11607881600925}a^{14}+\frac{2990135338358}{6964728960555}a^{13}-\frac{17492131108691}{34823644802775}a^{12}+\frac{28712253494719}{34823644802775}a^{11}+\frac{61467559213738}{34823644802775}a^{10}+\frac{635939382313}{6964728960555}a^{9}+\frac{52328774181428}{34823644802775}a^{8}+\frac{190503491228}{11607881600925}a^{7}-\frac{4905368582281}{34823644802775}a^{6}+\frac{92790353680399}{34823644802775}a^{5}+\frac{95589718120919}{34823644802775}a^{4}+\frac{234067550237}{3869293866975}a^{3}+\frac{61853751019048}{34823644802775}a^{2}-\frac{24229661189738}{11607881600925}a+\frac{33886752644122}{34823644802775}$, $\frac{6328482818033}{104470934408325}a^{22}-\frac{1609010200327}{34823644802775}a^{21}+\frac{90937286045051}{104470934408325}a^{20}-\frac{98104721866567}{104470934408325}a^{19}+\frac{49240804583179}{20894186881665}a^{18}-\frac{11053118349929}{104470934408325}a^{17}-\frac{27816089729183}{34823644802775}a^{16}+\frac{38384152967719}{20894186881665}a^{15}-\frac{532740946914004}{104470934408325}a^{14}+\frac{181731321882826}{104470934408325}a^{13}+\frac{766587522100483}{104470934408325}a^{12}-\frac{620810832209242}{104470934408325}a^{11}+\frac{88879050113164}{34823644802775}a^{10}-\frac{215854698168001}{34823644802775}a^{9}-\frac{178629109738673}{34823644802775}a^{8}+\frac{107455453023148}{11607881600925}a^{7}+\frac{985783034455189}{104470934408325}a^{6}-\frac{629585784862208}{104470934408325}a^{5}+\frac{53968560873479}{104470934408325}a^{4}-\frac{13\!\cdots\!94}{104470934408325}a^{3}+\frac{520158096087857}{34823644802775}a^{2}-\frac{148185856308536}{104470934408325}a-\frac{111995219393389}{104470934408325}$, $\frac{426946931497}{20894186881665}a^{22}+\frac{1857050648}{11607881600925}a^{21}+\frac{6337028802064}{20894186881665}a^{20}-\frac{9208425636181}{104470934408325}a^{19}+\frac{18106554079573}{20894186881665}a^{18}+\frac{10879207894607}{20894186881665}a^{17}+\frac{18443696688703}{34823644802775}a^{16}+\frac{120196599501722}{104470934408325}a^{15}-\frac{84099283223341}{104470934408325}a^{14}+\frac{52256996203567}{104470934408325}a^{13}+\frac{256262736542938}{104470934408325}a^{12}+\frac{18048686665283}{104470934408325}a^{11}+\frac{28297432563427}{11607881600925}a^{10}-\frac{34419578475127}{34823644802775}a^{9}-\frac{23529302043056}{11607881600925}a^{8}+\frac{48352587880987}{34823644802775}a^{7}+\frac{61455578373932}{20894186881665}a^{6}+\frac{83845005913876}{104470934408325}a^{5}+\frac{48362445348649}{20894186881665}a^{4}-\frac{440044108012124}{104470934408325}a^{3}+\frac{7645810337158}{3869293866975}a^{2}+\frac{46122078649846}{104470934408325}a+\frac{94828209208403}{104470934408325}$, $\frac{837076225127}{20894186881665}a^{22}-\frac{1531999244039}{34823644802775}a^{21}+\frac{12136515055157}{20894186881665}a^{20}-\frac{84708618406649}{104470934408325}a^{19}+\frac{35175214697867}{20894186881665}a^{18}-\frac{10752477313937}{20894186881665}a^{17}-\frac{29508895816163}{34823644802775}a^{16}+\frac{160182695427688}{104470934408325}a^{15}-\frac{427236561817169}{104470934408325}a^{14}+\frac{210414094328453}{104470934408325}a^{13}+\frac{544421032279202}{104470934408325}a^{12}-\frac{650800196747723}{104470934408325}a^{11}+\frac{12645693077936}{3869293866975}a^{10}-\frac{50259616708081}{11607881600925}a^{9}-\frac{141934065653477}{34823644802775}a^{8}+\frac{36117152256422}{3869293866975}a^{7}+\frac{95060506169671}{20894186881665}a^{6}-\frac{725781683012506}{104470934408325}a^{5}+\frac{55473120101228}{20894186881665}a^{4}-\frac{10\!\cdots\!66}{104470934408325}a^{3}+\frac{361699599663623}{34823644802775}a^{2}-\frac{26561769457231}{104470934408325}a-\frac{256873622113508}{104470934408325}$, $\frac{593165404982}{104470934408325}a^{22}-\frac{11547511786}{773858773395}a^{21}+\frac{8494903688369}{104470934408325}a^{20}-\frac{4989454072931}{20894186881665}a^{19}+\frac{5639767870888}{20894186881665}a^{18}-\frac{38172277643876}{104470934408325}a^{17}-\frac{3353122137902}{11607881600925}a^{16}+\frac{12297165517889}{104470934408325}a^{15}-\frac{66011405657443}{104470934408325}a^{14}+\frac{104980909477438}{104470934408325}a^{13}+\frac{90556170922453}{104470934408325}a^{12}-\frac{145046685991492}{104470934408325}a^{11}+\frac{15159380309908}{34823644802775}a^{10}-\frac{29508864818018}{34823644802775}a^{9}+\frac{25778655818702}{34823644802775}a^{8}+\frac{2648622825661}{1392945792111}a^{7}+\frac{40950630643876}{104470934408325}a^{6}-\frac{26507751894538}{20894186881665}a^{5}-\frac{43221530068504}{104470934408325}a^{4}-\frac{336955089298469}{104470934408325}a^{3}+\frac{115514233353967}{34823644802775}a^{2}-\frac{11363505136652}{104470934408325}a-\frac{14541301408868}{20894186881665}$, $\frac{510737237161}{104470934408325}a^{22}-\frac{9882716008}{34823644802775}a^{21}+\frac{6849542042857}{104470934408325}a^{20}-\frac{3242429795603}{104470934408325}a^{19}+\frac{2284843336649}{20894186881665}a^{18}+\frac{6411566791832}{104470934408325}a^{17}-\frac{1031894126258}{11607881600925}a^{16}-\frac{20614730570762}{104470934408325}a^{15}-\frac{33339024020312}{104470934408325}a^{14}-\frac{6607379079191}{20894186881665}a^{13}+\frac{43522094596883}{104470934408325}a^{12}+\frac{82595478077443}{104470934408325}a^{11}-\frac{14695468671941}{11607881600925}a^{10}+\frac{3073880747659}{6964728960555}a^{9}-\frac{58291904948638}{34823644802775}a^{8}+\frac{35549506381081}{34823644802775}a^{7}+\frac{161259754324913}{104470934408325}a^{6}+\frac{161701349983553}{104470934408325}a^{5}-\frac{206490197772107}{104470934408325}a^{4}+\frac{38452162727636}{104470934408325}a^{3}-\frac{61772555300693}{34823644802775}a^{2}+\frac{209900478601922}{104470934408325}a-\frac{70458919710326}{104470934408325}$, $\frac{9324208177}{159012076725}a^{22}-\frac{45941699659}{477036230175}a^{21}+\frac{425620134937}{477036230175}a^{20}-\frac{265029910346}{159012076725}a^{19}+\frac{11403838379}{3533601705}a^{18}-\frac{1132668657013}{477036230175}a^{17}-\frac{127303413232}{477036230175}a^{16}+\frac{127105096637}{53004025575}a^{15}-\frac{3363790676852}{477036230175}a^{14}+\frac{647582276972}{95407246035}a^{13}+\frac{2362018550128}{477036230175}a^{12}-\frac{5650310202107}{477036230175}a^{11}+\frac{4618954084856}{477036230175}a^{10}-\frac{1079628038279}{95407246035}a^{9}-\frac{50428555738}{159012076725}a^{8}+\frac{7234985400143}{477036230175}a^{7}-\frac{356823870422}{477036230175}a^{6}-\frac{5545770348592}{477036230175}a^{5}+\frac{1265903807081}{159012076725}a^{4}-\frac{9202302183359}{477036230175}a^{3}+\frac{1395572284544}{53004025575}a^{2}-\frac{5026813939868}{477036230175}a+\frac{1951810228}{159012076725}$, $a$ Copy content Toggle raw display (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:  \( 530761705.173 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 530761705.173 \cdot 1}{2\cdot\sqrt{58230945563524325903440619509366903}}\cr\approx \mathstrut & 1.32526037075 \end{aligned}\] (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 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*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 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*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 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*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 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*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 $23$ ${\href{/padicField/3.2.0.1}{2} }^{11}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{11}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ $23$ ${\href{/padicField/29.2.0.1}{2} }^{11}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $23$ $23$ ${\href{/padicField/41.2.0.1}{2} }^{11}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $23$ ${\href{/padicField/47.2.0.1}{2} }^{11}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $23$ $23$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(1447\) Copy content Toggle raw display $\Q_{1447}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
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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}$