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Properties

Label 43.43.162...801.1
Degree $43$
Signature $(43, 0)$
Discriminant $1.627\times 10^{137}$
Root discriminant \(1552.25\)
Ramified prime $43$
Class number not computed
Class group not computed
Galois group $C_{43}$ (as 43T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727)
 
Copy content gp:K = bnfinit(y^43 - 903*y^41 - 645*y^40 + 358276*y^39 + 487362*y^38 - 82668446*y^37 - 161379172*y^36 + 12386889944*y^35 + 31035129471*y^34 - 1275092913940*y^33 - 3868686195114*y^32 + 93052767268909*y^31 + 330473143424644*y^30 - 4897798025442393*y^29 - 19967338988758228*y^28 + 187329847291648022*y^27 + 869728040966016241*y^26 - 5201925841676982029*y^25 - 27621776964130665961*y^24 + 103748981858304966660*y^23 + 643326429466437698421*y^22 - 1445787294046167541816*y^21 - 10999588529245966977339*y^20 + 13140703542377826450494*y^19 + 137573117593988043652238*y^18 - 60838518194479293931722*y^17 - 1247516677692517201034941*y^16 - 134666416459554320685647*y^15 + 8070410654031859605939559*y^14 + 4203376282667672263555078*y^13 - 36228120784944419229616219*y^12 - 30515963297503545387850753*y^11 + 107534691527795878017440861*y^10 + 119306453339061961678830425*y^9 - 192818992256047972290970619*y^8 - 263326021640166500214931210*y^7 + 170563929616560831037152772*y^6 + 294222510747058121054570076*y^5 - 33622183448420036732904682*y^4 - 118832831442545361067982885*y^3 - 7933826886293481520543865*y^2 + 14571491290875996179048369*y + 2572343484535669027372727, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727)
 

\( x^{43} - 903 x^{41} - 645 x^{40} + 358276 x^{39} + 487362 x^{38} - 82668446 x^{37} + \cdots + 25\!\cdots\!27 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $43$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(43, 0)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(162\!\cdots\!801\) \(\medspace = 43^{84}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(1552.25\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $43^{84/43}\approx 1552.2498771818027$
Ramified primes:   \(43\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{43}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1849=43^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1849}(1,·)$, $\chi_{1849}(130,·)$, $\chi_{1849}(259,·)$, $\chi_{1849}(388,·)$, $\chi_{1849}(517,·)$, $\chi_{1849}(646,·)$, $\chi_{1849}(775,·)$, $\chi_{1849}(904,·)$, $\chi_{1849}(1033,·)$, $\chi_{1849}(1162,·)$, $\chi_{1849}(1291,·)$, $\chi_{1849}(1420,·)$, $\chi_{1849}(1549,·)$, $\chi_{1849}(1678,·)$, $\chi_{1849}(1807,·)$, $\chi_{1849}(44,·)$, $\chi_{1849}(173,·)$, $\chi_{1849}(302,·)$, $\chi_{1849}(431,·)$, $\chi_{1849}(560,·)$, $\chi_{1849}(689,·)$, $\chi_{1849}(818,·)$, $\chi_{1849}(947,·)$, $\chi_{1849}(1076,·)$, $\chi_{1849}(1205,·)$, $\chi_{1849}(1334,·)$, $\chi_{1849}(1463,·)$, $\chi_{1849}(1592,·)$, $\chi_{1849}(1721,·)$, $\chi_{1849}(87,·)$, $\chi_{1849}(216,·)$, $\chi_{1849}(345,·)$, $\chi_{1849}(474,·)$, $\chi_{1849}(603,·)$, $\chi_{1849}(732,·)$, $\chi_{1849}(861,·)$, $\chi_{1849}(990,·)$, $\chi_{1849}(1119,·)$, $\chi_{1849}(1248,·)$, $\chi_{1849}(1377,·)$, $\chi_{1849}(1506,·)$, $\chi_{1849}(1635,·)$, $\chi_{1849}(1764,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{19}a^{19}-\frac{1}{19}a$, $\frac{1}{19}a^{20}-\frac{1}{19}a^{2}$, $\frac{1}{19}a^{21}-\frac{1}{19}a^{3}$, $\frac{1}{19}a^{22}-\frac{1}{19}a^{4}$, $\frac{1}{19}a^{23}-\frac{1}{19}a^{5}$, $\frac{1}{19}a^{24}-\frac{1}{19}a^{6}$, $\frac{1}{19}a^{25}-\frac{1}{19}a^{7}$, $\frac{1}{19}a^{26}-\frac{1}{19}a^{8}$, $\frac{1}{19}a^{27}-\frac{1}{19}a^{9}$, $\frac{1}{19}a^{28}-\frac{1}{19}a^{10}$, $\frac{1}{19}a^{29}-\frac{1}{19}a^{11}$, $\frac{1}{19}a^{30}-\frac{1}{19}a^{12}$, $\frac{1}{361}a^{31}-\frac{1}{361}a^{30}+\frac{9}{361}a^{29}+\frac{3}{361}a^{28}+\frac{7}{361}a^{27}+\frac{2}{361}a^{26}+\frac{1}{361}a^{25}+\frac{4}{361}a^{24}+\frac{5}{361}a^{23}-\frac{1}{361}a^{22}-\frac{4}{361}a^{21}-\frac{7}{361}a^{20}+\frac{3}{19}a^{18}-\frac{5}{19}a^{16}-\frac{8}{19}a^{15}-\frac{7}{19}a^{14}-\frac{134}{361}a^{13}+\frac{172}{361}a^{12}-\frac{180}{361}a^{11}-\frac{79}{361}a^{10}-\frac{83}{361}a^{9}-\frac{78}{361}a^{8}-\frac{115}{361}a^{7}-\frac{61}{361}a^{6}-\frac{119}{361}a^{5}-\frac{75}{361}a^{4}+\frac{42}{361}a^{3}-\frac{69}{361}a^{2}$, $\frac{1}{361}a^{32}+\frac{8}{361}a^{30}-\frac{7}{361}a^{29}-\frac{9}{361}a^{28}+\frac{9}{361}a^{27}+\frac{3}{361}a^{26}+\frac{5}{361}a^{25}+\frac{9}{361}a^{24}+\frac{4}{361}a^{23}-\frac{5}{361}a^{22}+\frac{8}{361}a^{21}-\frac{7}{361}a^{20}+\frac{3}{19}a^{18}-\frac{5}{19}a^{17}+\frac{6}{19}a^{16}+\frac{4}{19}a^{15}+\frac{94}{361}a^{14}+\frac{2}{19}a^{13}-\frac{8}{361}a^{12}+\frac{121}{361}a^{11}-\frac{143}{361}a^{10}-\frac{161}{361}a^{9}+\frac{168}{361}a^{8}-\frac{176}{361}a^{7}-\frac{180}{361}a^{6}+\frac{167}{361}a^{5}-\frac{33}{361}a^{4}-\frac{46}{361}a^{3}-\frac{69}{361}a^{2}+\frac{3}{19}a$, $\frac{1}{361}a^{33}+\frac{1}{361}a^{30}-\frac{5}{361}a^{29}+\frac{4}{361}a^{28}+\frac{4}{361}a^{27}+\frac{8}{361}a^{26}+\frac{1}{361}a^{25}-\frac{9}{361}a^{24}-\frac{7}{361}a^{23}-\frac{3}{361}a^{22}+\frac{6}{361}a^{21}-\frac{1}{361}a^{20}+\frac{9}{19}a^{18}+\frac{6}{19}a^{17}+\frac{6}{19}a^{16}-\frac{134}{361}a^{15}+\frac{1}{19}a^{14}-\frac{1}{19}a^{13}-\frac{172}{361}a^{12}+\frac{138}{361}a^{11}+\frac{91}{361}a^{10}+\frac{53}{361}a^{9}+\frac{68}{361}a^{8}+\frac{18}{361}a^{7}-\frac{86}{361}a^{6}+\frac{159}{361}a^{5}-\frac{149}{361}a^{4}-\frac{25}{361}a^{3}-\frac{56}{361}a^{2}+\frac{3}{19}a$, $\frac{1}{361}a^{34}-\frac{4}{361}a^{30}-\frac{5}{361}a^{29}+\frac{1}{361}a^{28}+\frac{1}{361}a^{27}-\frac{1}{361}a^{26}+\frac{9}{361}a^{25}+\frac{8}{361}a^{24}-\frac{8}{361}a^{23}+\frac{7}{361}a^{22}+\frac{3}{361}a^{21}+\frac{7}{361}a^{20}+\frac{3}{19}a^{18}+\frac{6}{19}a^{17}-\frac{39}{361}a^{16}+\frac{9}{19}a^{15}+\frac{6}{19}a^{14}-\frac{2}{19}a^{13}-\frac{34}{361}a^{12}-\frac{90}{361}a^{11}+\frac{132}{361}a^{10}+\frac{151}{361}a^{9}+\frac{96}{361}a^{8}+\frac{10}{361}a^{7}-\frac{160}{361}a^{6}-\frac{30}{361}a^{5}+\frac{50}{361}a^{4}-\frac{98}{361}a^{3}+\frac{126}{361}a^{2}+\frac{9}{19}a$, $\frac{1}{361}a^{35}-\frac{9}{361}a^{30}-\frac{1}{361}a^{29}-\frac{6}{361}a^{28}+\frac{8}{361}a^{27}-\frac{2}{361}a^{26}-\frac{7}{361}a^{25}+\frac{8}{361}a^{24}+\frac{8}{361}a^{23}-\frac{1}{361}a^{22}-\frac{9}{361}a^{21}-\frac{9}{361}a^{20}-\frac{1}{19}a^{18}-\frac{39}{361}a^{17}+\frac{8}{19}a^{16}-\frac{7}{19}a^{15}+\frac{8}{19}a^{14}+\frac{8}{19}a^{13}-\frac{124}{361}a^{12}+\frac{172}{361}a^{11}-\frac{146}{361}a^{10}+\frac{144}{361}a^{9}+\frac{78}{361}a^{8}+\frac{121}{361}a^{7}+\frac{87}{361}a^{6}-\frac{46}{361}a^{5}-\frac{37}{361}a^{4}-\frac{67}{361}a^{3}-\frac{124}{361}a^{2}+\frac{3}{19}a$, $\frac{1}{361}a^{36}+\frac{9}{361}a^{30}-\frac{1}{361}a^{29}-\frac{3}{361}a^{28}+\frac{4}{361}a^{27}-\frac{8}{361}a^{26}-\frac{2}{361}a^{25}+\frac{6}{361}a^{24}+\frac{6}{361}a^{23}+\frac{1}{361}a^{22}-\frac{7}{361}a^{21}-\frac{6}{361}a^{20}+\frac{113}{361}a^{18}+\frac{8}{19}a^{17}+\frac{5}{19}a^{16}-\frac{7}{19}a^{15}+\frac{2}{19}a^{14}+\frac{6}{19}a^{13}-\frac{104}{361}a^{12}+\frac{115}{361}a^{11}-\frac{168}{361}a^{10}+\frac{110}{361}a^{9}+\frac{160}{361}a^{8}+\frac{154}{361}a^{7}+\frac{165}{361}a^{6}+\frac{13}{361}a^{5}-\frac{39}{361}a^{4}-\frac{145}{361}a^{3}+\frac{101}{361}a^{2}-\frac{1}{19}a$, $\frac{1}{361}a^{37}+\frac{8}{361}a^{30}-\frac{8}{361}a^{29}-\frac{4}{361}a^{28}+\frac{5}{361}a^{27}-\frac{1}{361}a^{26}-\frac{3}{361}a^{25}+\frac{8}{361}a^{24}-\frac{6}{361}a^{23}+\frac{2}{361}a^{22}-\frac{8}{361}a^{21}+\frac{6}{361}a^{20}-\frac{1}{361}a^{19}+\frac{5}{19}a^{17}-\frac{2}{19}a^{15}-\frac{7}{19}a^{14}+\frac{1}{19}a^{13}+\frac{11}{361}a^{12}-\frac{68}{361}a^{11}+\frac{80}{361}a^{10}+\frac{109}{361}a^{9}+\frac{115}{361}a^{8}+\frac{117}{361}a^{7}+\frac{163}{361}a^{6}-\frac{89}{361}a^{5}+\frac{169}{361}a^{4}+\frac{122}{361}a^{3}-\frac{63}{361}a^{2}+\frac{6}{19}a$, $\frac{1}{6859}a^{38}+\frac{5}{6859}a^{37}-\frac{5}{6859}a^{36}+\frac{4}{6859}a^{35}-\frac{5}{6859}a^{34}-\frac{1}{6859}a^{33}-\frac{8}{6859}a^{32}+\frac{1}{6859}a^{31}-\frac{163}{6859}a^{30}-\frac{1}{6859}a^{29}-\frac{115}{6859}a^{28}+\frac{77}{6859}a^{27}+\frac{173}{6859}a^{26}+\frac{91}{6859}a^{25}-\frac{7}{361}a^{24}+\frac{68}{6859}a^{23}-\frac{30}{6859}a^{22}+\frac{174}{6859}a^{21}-\frac{134}{6859}a^{20}-\frac{119}{6859}a^{19}-\frac{1857}{6859}a^{18}-\frac{726}{6859}a^{17}-\frac{793}{6859}a^{16}-\frac{2260}{6859}a^{15}+\frac{445}{6859}a^{14}+\frac{626}{6859}a^{13}+\frac{3260}{6859}a^{12}-\frac{1728}{6859}a^{11}-\frac{2849}{6859}a^{10}+\frac{854}{6859}a^{9}+\frac{2639}{6859}a^{8}-\frac{851}{6859}a^{7}-\frac{144}{361}a^{6}-\frac{1151}{6859}a^{5}-\frac{2231}{6859}a^{4}-\frac{3005}{6859}a^{3}-\frac{141}{361}a^{2}+\frac{2}{19}a$, $\frac{1}{6859}a^{39}+\frac{8}{6859}a^{37}-\frac{9}{6859}a^{36}-\frac{6}{6859}a^{35}+\frac{5}{6859}a^{34}-\frac{3}{6859}a^{33}+\frac{3}{6859}a^{32}+\frac{3}{6859}a^{31}-\frac{155}{6859}a^{30}+\frac{61}{6859}a^{29}-\frac{108}{6859}a^{28}+\frac{92}{6859}a^{27}+\frac{62}{6859}a^{26}+\frac{134}{6859}a^{25}+\frac{68}{6859}a^{24}-\frac{180}{6859}a^{23}-\frac{132}{6859}a^{22}-\frac{92}{6859}a^{21}+\frac{7}{361}a^{20}+\frac{144}{6859}a^{19}-\frac{3316}{6859}a^{18}+\frac{1374}{6859}a^{17}+\frac{1724}{6859}a^{16}+\frac{1276}{6859}a^{15}+\frac{605}{6859}a^{14}+\frac{3208}{6859}a^{13}+\frac{991}{6859}a^{12}-\frac{2683}{6859}a^{11}-\frac{1108}{6859}a^{10}-\frac{2296}{6859}a^{9}-\frac{1886}{6859}a^{8}+\frac{1880}{6859}a^{7}+\frac{920}{6859}a^{6}+\frac{446}{6859}a^{5}+\frac{3191}{6859}a^{4}+\frac{2409}{6859}a^{3}+\frac{24}{361}a^{2}+\frac{7}{19}a$, $\frac{1}{6859}a^{40}+\frac{8}{6859}a^{37}-\frac{4}{6859}a^{36}-\frac{8}{6859}a^{35}-\frac{1}{6859}a^{34}-\frac{8}{6859}a^{33}-\frac{9}{6859}a^{32}+\frac{8}{6859}a^{31}-\frac{60}{6859}a^{30}+\frac{14}{6859}a^{29}+\frac{62}{6859}a^{28}+\frac{130}{6859}a^{27}+\frac{42}{6859}a^{26}-\frac{14}{6859}a^{25}+\frac{48}{6859}a^{24}-\frac{106}{6859}a^{23}-\frac{156}{6859}a^{22}+\frac{109}{6859}a^{21}-\frac{1}{361}a^{20}+\frac{106}{6859}a^{19}-\frac{2143}{6859}a^{18}+\frac{293}{6859}a^{17}+\frac{1882}{6859}a^{16}-\frac{2956}{6859}a^{15}+\frac{85}{6859}a^{14}-\frac{939}{6859}a^{13}+\frac{1903}{6859}a^{12}-\frac{2199}{6859}a^{11}+\frac{508}{6859}a^{10}+\frac{1067}{6859}a^{9}-\frac{1391}{6859}a^{8}+\frac{2389}{6859}a^{7}-\frac{2100}{6859}a^{6}-\frac{1528}{6859}a^{5}+\frac{1067}{6859}a^{4}-\frac{698}{6859}a^{3}-\frac{118}{361}a^{2}+\frac{5}{19}a$, $\frac{1}{6878015143969}a^{41}+\frac{435859134}{6878015143969}a^{40}+\frac{85763745}{6878015143969}a^{39}+\frac{45050261}{6878015143969}a^{38}+\frac{7927350419}{6878015143969}a^{37}-\frac{237177735}{6878015143969}a^{36}+\frac{1113788179}{6878015143969}a^{35}-\frac{275569336}{362000797051}a^{34}-\frac{9214885919}{6878015143969}a^{33}-\frac{433674781}{362000797051}a^{32}-\frac{1326333093}{6878015143969}a^{31}-\frac{56832712826}{6878015143969}a^{30}+\frac{89881617977}{6878015143969}a^{29}-\frac{46926275052}{6878015143969}a^{28}-\frac{61062312690}{6878015143969}a^{27}-\frac{137765568174}{6878015143969}a^{26}+\frac{140250252658}{6878015143969}a^{25}+\frac{43312967627}{6878015143969}a^{24}+\frac{141123855759}{6878015143969}a^{23}+\frac{62011859918}{6878015143969}a^{22}+\frac{67626050954}{6878015143969}a^{21}-\frac{133575666681}{6878015143969}a^{20}-\frac{58182045108}{6878015143969}a^{19}-\frac{1935685410316}{6878015143969}a^{18}-\frac{2332798217041}{6878015143969}a^{17}+\frac{123022095597}{362000797051}a^{16}-\frac{2625605664946}{6878015143969}a^{15}-\frac{12945019894}{362000797051}a^{14}+\frac{120576757677}{6878015143969}a^{13}-\frac{652815817980}{6878015143969}a^{12}+\frac{1790924808256}{6878015143969}a^{11}-\frac{2054956186096}{6878015143969}a^{10}-\frac{616489071502}{6878015143969}a^{9}+\frac{411196198431}{6878015143969}a^{8}-\frac{82126745323}{6878015143969}a^{7}+\frac{1651430747875}{6878015143969}a^{6}-\frac{1126766198333}{6878015143969}a^{5}+\frac{1834872080832}{6878015143969}a^{4}-\frac{89062323636}{362000797051}a^{3}+\frac{4698337468}{19052673529}a^{2}+\frac{74528819}{1002772291}a-\frac{1693977}{52777489}$, $\frac{1}{45\cdots 83}a^{42}+\frac{16\cdots 31}{45\cdots 83}a^{41}+\frac{17\cdots 46}{45\cdots 83}a^{40}-\frac{21\cdots 15}{45\cdots 83}a^{39}-\frac{30\cdots 12}{45\cdots 83}a^{38}+\frac{37\cdots 43}{45\cdots 83}a^{37}+\frac{24\cdots 48}{45\cdots 83}a^{36}+\frac{56\cdots 76}{45\cdots 83}a^{35}-\frac{47\cdots 72}{45\cdots 83}a^{34}-\frac{49\cdots 96}{45\cdots 83}a^{33}+\frac{14\cdots 98}{45\cdots 83}a^{32}-\frac{45\cdots 99}{45\cdots 83}a^{31}-\frac{41\cdots 40}{24\cdots 57}a^{30}-\frac{12\cdots 62}{45\cdots 83}a^{29}-\frac{36\cdots 63}{45\cdots 83}a^{28}-\frac{37\cdots 85}{45\cdots 83}a^{27}+\frac{64\cdots 37}{45\cdots 83}a^{26}+\frac{11\cdots 66}{45\cdots 83}a^{25}+\frac{11\cdots 05}{45\cdots 83}a^{24}+\frac{11\cdots 07}{45\cdots 83}a^{23}-\frac{28\cdots 81}{45\cdots 83}a^{22}-\frac{18\cdots 63}{45\cdots 83}a^{21}-\frac{81\cdots 11}{45\cdots 83}a^{20}+\frac{10\cdots 64}{45\cdots 83}a^{19}-\frac{11\cdots 70}{45\cdots 83}a^{18}+\frac{21\cdots 59}{45\cdots 83}a^{17}-\frac{15\cdots 18}{45\cdots 83}a^{16}+\frac{21\cdots 30}{45\cdots 83}a^{15}-\frac{19\cdots 85}{45\cdots 83}a^{14}+\frac{12\cdots 67}{45\cdots 83}a^{13}-\frac{89\cdots 23}{24\cdots 57}a^{12}+\frac{15\cdots 91}{45\cdots 83}a^{11}-\frac{22\cdots 57}{45\cdots 83}a^{10}+\frac{84\cdots 77}{45\cdots 83}a^{9}-\frac{22\cdots 80}{45\cdots 83}a^{8}+\frac{69\cdots 43}{45\cdots 83}a^{7}+\frac{20\cdots 20}{45\cdots 83}a^{6}-\frac{43\cdots 25}{45\cdots 83}a^{5}-\frac{18\cdots 79}{45\cdots 83}a^{4}-\frac{29\cdots 99}{24\cdots 57}a^{3}+\frac{27\cdots 24}{12\cdots 03}a^{2}-\frac{24\cdots 89}{66\cdots 37}a-\frac{16\cdots 80}{35\cdots 23}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $19$

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $42$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{43}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{162686032778208990102858628859785420567496242104134005559503199497609643882923419981647276367075859293620549051195773051892887390454194801}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727); 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

$C_{43}$ (as 43T1):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A cyclic group of order 43
The 43 conjugacy class representatives for $C_{43}$
Character table for $C_{43}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $43$ $43$ $43$ $43$ $43$ $43$ $43$ ${\href{/padicField/19.1.0.1}{1} }^{43}$ $43$ $43$ $43$ $43$ $43$ R $43$ $43$ $43$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(43\) Copy content Toggle raw display Deg $43$$43$$1$$84$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)