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Properties

Label 40.0.102...625.1
Degree $40$
Signature $[0, 20]$
Discriminant $1.028\times 10^{61}$
Root discriminant \(33.52\)
Ramified primes $3,5,11$
Class number $22$ (GRH)
Class group [22] (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 
gp: K = bnfinit(y^40 - y^39 - y^38 + 4*y^37 - 4*y^36 - 4*y^35 + 17*y^34 - 17*y^33 - 17*y^32 + 72*y^31 - 72*y^30 + 127*y^29 + 106*y^28 - 504*y^27 + 491*y^26 + 496*y^25 - 2088*y^24 + 2091*y^23 + 2090*y^22 - 8856*y^21 + 8855*y^20 + 8856*y^19 + 2090*y^18 - 2091*y^17 - 2088*y^16 - 496*y^15 + 491*y^14 + 504*y^13 + 106*y^12 - 127*y^11 - 72*y^10 - 72*y^9 - 17*y^8 + 17*y^7 + 17*y^6 + 4*y^5 - 4*y^4 - 4*y^3 - y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 

\( x^{40} - x^{39} - x^{38} + 4 x^{37} - 4 x^{36} - 4 x^{35} + 17 x^{34} - 17 x^{33} - 17 x^{32} + 72 x^{31} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(10279259898673041257519092860384627329993230987644195556640625\) \(\medspace = 3^{20}\cdot 5^{20}\cdot 11^{36}\) 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:  \(33.52\)
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:  $3^{1/2}5^{1/2}11^{9/10}\approx 33.51961688386638$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(134,·)$, $\chi_{165}(136,·)$, $\chi_{165}(139,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(146,·)$, $\chi_{165}(19,·)$, $\chi_{165}(149,·)$, $\chi_{165}(151,·)$, $\chi_{165}(26,·)$, $\chi_{165}(29,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(164,·)$, $\chi_{165}(41,·)$, $\chi_{165}(46,·)$, $\chi_{165}(49,·)$, $\chi_{165}(56,·)$, $\chi_{165}(59,·)$, $\chi_{165}(61,·)$, $\chi_{165}(64,·)$, $\chi_{165}(71,·)$, $\chi_{165}(74,·)$, $\chi_{165}(76,·)$, $\chi_{165}(79,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(94,·)$, $\chi_{165}(101,·)$, $\chi_{165}(104,·)$, $\chi_{165}(106,·)$, $\chi_{165}(109,·)$, $\chi_{165}(116,·)$, $\chi_{165}(119,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{89}a^{22}-\frac{34}{89}a^{11}-\frac{1}{89}$, $\frac{1}{89}a^{23}-\frac{34}{89}a^{12}-\frac{1}{89}a$, $\frac{1}{89}a^{24}-\frac{34}{89}a^{13}-\frac{1}{89}a^{2}$, $\frac{1}{89}a^{25}-\frac{34}{89}a^{14}-\frac{1}{89}a^{3}$, $\frac{1}{89}a^{26}-\frac{34}{89}a^{15}-\frac{1}{89}a^{4}$, $\frac{1}{89}a^{27}-\frac{34}{89}a^{16}-\frac{1}{89}a^{5}$, $\frac{1}{89}a^{28}-\frac{34}{89}a^{17}-\frac{1}{89}a^{6}$, $\frac{1}{89}a^{29}-\frac{34}{89}a^{18}-\frac{1}{89}a^{7}$, $\frac{1}{178}a^{30}-\frac{1}{178}a^{27}-\frac{1}{178}a^{24}-\frac{1}{2}a^{21}-\frac{17}{89}a^{19}-\frac{1}{2}a^{18}+\frac{17}{89}a^{16}-\frac{1}{2}a^{15}+\frac{17}{89}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}+\frac{44}{89}a^{8}-\frac{1}{2}a^{6}-\frac{44}{89}a^{5}-\frac{1}{2}a^{3}-\frac{44}{89}a^{2}-\frac{1}{2}$, $\frac{1}{3524578}a^{31}-\frac{2338}{1762289}a^{30}-\frac{7564}{1762289}a^{29}-\frac{15121}{3524578}a^{28}+\frac{549}{1762289}a^{27}+\frac{9346}{1762289}a^{26}+\frac{18721}{3524578}a^{25}-\frac{142}{1762289}a^{24}-\frac{9782}{1762289}a^{23}-\frac{19441}{3524578}a^{22}+\frac{3782}{19801}a^{21}-\frac{544608}{1762289}a^{20}-\frac{930287}{3524578}a^{19}-\frac{79422}{1762289}a^{18}+\frac{801737}{1762289}a^{17}+\frac{219611}{3524578}a^{16}+\frac{434731}{1762289}a^{15}-\frac{446417}{1762289}a^{14}-\frac{51843}{3524578}a^{13}-\frac{518608}{1762289}a^{12}+\frac{362537}{1762289}a^{11}-\frac{19441}{39602}a^{10}+\frac{3782}{1762289}a^{9}-\frac{45722}{1762289}a^{8}-\frac{1666171}{3524578}a^{7}+\frac{50948}{1762289}a^{6}-\frac{176769}{1762289}a^{5}-\frac{1424981}{3524578}a^{4}+\frac{167972}{1762289}a^{3}-\frac{752798}{1762289}a^{2}-\frac{237735}{3524578}a+\frac{762438}{1762289}$, $\frac{1}{3524578}a^{32}+\frac{1}{3524578}a^{30}+\frac{15125}{3524578}a^{29}-\frac{7564}{1762289}a^{28}-\frac{15121}{3524578}a^{27}-\frac{18703}{3524578}a^{26}+\frac{9346}{1762289}a^{25}+\frac{18721}{3524578}a^{24}+\frac{19517}{3524578}a^{23}-\frac{9782}{1762289}a^{22}+\frac{1089415}{3524578}a^{21}-\frac{12237}{39602}a^{20}-\frac{544608}{1762289}a^{19}-\frac{1603521}{3524578}a^{18}+\frac{1603445}{3524578}a^{17}+\frac{801737}{1762289}a^{16}+\frac{892845}{3524578}a^{15}-\frac{892827}{3524578}a^{14}-\frac{446417}{1762289}a^{13}-\frac{725077}{3524578}a^{12}+\frac{725073}{3524578}a^{11}-\frac{865125}{1762289}a^{10}-\frac{19441}{39602}a^{9}-\frac{1754725}{3524578}a^{8}+\frac{825522}{1762289}a^{7}-\frac{1666171}{3524578}a^{6}-\frac{1660393}{3524578}a^{5}+\frac{714276}{1762289}a^{4}-\frac{1424981}{3524578}a^{3}-\frac{1426345}{3524578}a^{2}+\frac{118446}{1762289}a-\frac{237735}{3524578}$, $\frac{1}{3524578}a^{33}+\frac{1346269}{3524578}$, $\frac{1}{3524578}a^{34}+\frac{1346269}{3524578}a$, $\frac{1}{3524578}a^{35}+\frac{1346269}{3524578}a^{2}$, $\frac{1}{3524578}a^{36}+\frac{1346269}{3524578}a^{3}$, $\frac{1}{3524578}a^{37}+\frac{1346269}{3524578}a^{4}$, $\frac{1}{3524578}a^{38}+\frac{1346269}{3524578}a^{5}$, $\frac{1}{3524578}a^{39}+\frac{1346269}{3524578}a^{6}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{22}$, which has order $22$ (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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{3}{3524578} a^{37} - \frac{24157817}{3524578} a^{4} \)  (order $66$) 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:   $\frac{21}{3524578}a^{39}-\frac{21}{1762289}a^{38}+\frac{105}{3524578}a^{36}-\frac{84}{1762289}a^{35}+\frac{441}{3524578}a^{33}-\frac{357}{1762289}a^{32}+\frac{4179}{3524578}a^{31}+\frac{21}{39602}a^{30}-\frac{1512}{1762289}a^{29}+\frac{4179}{3524578}a^{28}-\frac{441}{3524578}a^{27}-\frac{6405}{1762289}a^{26}+\frac{20895}{3524578}a^{25}+\frac{105}{3524578}a^{24}-\frac{27132}{1762289}a^{23}+\frac{87759}{3524578}a^{22}-\frac{21}{3524578}a^{21}+\frac{300888}{1762289}a^{20}+\frac{4179}{39602}a^{19}+\frac{21}{3524578}a^{18}-\frac{71043}{1762289}a^{17}-\frac{87801}{3524578}a^{16}+\frac{63}{3524578}a^{15}+\frac{16716}{1762289}a^{14}+\frac{20727}{3524578}a^{13}+\frac{273}{3524578}a^{12}-\frac{4179}{1762289}a^{11}-\frac{4893}{3524578}a^{10}+\frac{165577117}{3524578}a^{9}+\frac{1155}{3524578}a^{7}+\frac{357}{1762289}a^{6}-\frac{273}{3524578}a^{4}-\frac{84}{1762289}a^{3}+\frac{63}{3524578}a+\frac{21}{1762289}$, $\frac{257091}{1762289}a^{39}+\frac{257142}{1762289}a^{38}-\frac{1028432}{1762289}a^{37}+\frac{1028347}{1762289}a^{36}+\frac{1028568}{1762289}a^{35}-\frac{4370836}{1762289}a^{34}+\frac{4370479}{1762289}a^{33}+\frac{4371414}{1762289}a^{32}-\frac{18515159}{1762289}a^{31}+\frac{18510263}{1762289}a^{30}+\frac{18515516}{1762289}a^{29}-\frac{27256831}{1762289}a^{28}+\frac{129582789}{1762289}a^{27}-\frac{126229658}{1762289}a^{26}-\frac{127542483}{1762289}a^{25}+\frac{536841419}{1762289}a^{24}-\frac{6040100}{19801}a^{23}-\frac{537426763}{1762289}a^{22}+\frac{2276948465}{1762289}a^{21}-\frac{2277178492}{1762289}a^{20}-\frac{2277249535}{1762289}a^{19}+\frac{9644892692}{1762289}a^{18}+\frac{537727850}{1762289}a^{17}+\frac{536912581}{1762289}a^{16}+\frac{127525517}{1762289}a^{15}-\frac{126267092}{1762289}a^{14}-\frac{129599211}{1762289}a^{13}-\frac{306221}{19801}a^{12}+\frac{32659482}{1762289}a^{11}+\frac{18515737}{1762289}a^{10}-\frac{115442924}{1762289}a^{9}+\frac{4370836}{1762289}a^{8}-\frac{4370479}{1762289}a^{7}-\frac{4371414}{1762289}a^{6}-\frac{1028432}{1762289}a^{5}+\frac{1028653}{1762289}a^{4}+\frac{1028568}{1762289}a^{3}+\frac{257108}{1762289}a^{2}-\frac{257159}{1762289}a-\frac{257142}{1762289}$, $\frac{23184}{1762289}a^{35}+\frac{14328}{1762289}a^{34}+\frac{233}{89}a^{24}+\frac{144}{89}a^{23}+\frac{46368}{89}a^{13}+\frac{28657}{89}a^{12}-\frac{4613616}{1762289}a^{2}+\frac{72}{1762289}a$, $\frac{4}{1762289}a^{39}-\frac{14328}{1762289}a^{34}-\frac{144}{89}a^{23}-\frac{28657}{89}a^{12}-\frac{31622993}{1762289}a^{6}-\frac{72}{1762289}a-1$, $\frac{257108}{1762289}a^{39}+\frac{257108}{1762289}a^{38}-\frac{1028432}{1762289}a^{37}+\frac{1028432}{1762289}a^{36}+\frac{1028432}{1762289}a^{35}-\frac{4370836}{1762289}a^{34}+\frac{4370836}{1762289}a^{33}+\frac{4370836}{1762289}a^{32}-\frac{18511776}{1762289}a^{31}+\frac{18511776}{1762289}a^{30}+\frac{208012}{19801}a^{29}-\frac{27253448}{1762289}a^{28}+\frac{129582432}{1762289}a^{27}-\frac{126240028}{1762289}a^{26}-\frac{127525568}{1762289}a^{25}+\frac{536841504}{1762289}a^{24}-\frac{537612828}{1762289}a^{23}-\frac{537355720}{1762289}a^{22}+\frac{2276948448}{1762289}a^{21}-\frac{2276691340}{1762289}a^{20}-\frac{2276948448}{1762289}a^{19}+\frac{108369581}{19801}a^{18}+\frac{537612828}{1762289}a^{17}+\frac{536841504}{1762289}a^{16}+\frac{127525568}{1762289}a^{15}-\frac{126240028}{1762289}a^{14}-\frac{129582432}{1762289}a^{13}-\frac{27253448}{1762289}a^{12}+\frac{32652716}{1762289}a^{11}+\frac{18511776}{1762289}a^{10}+\frac{18511776}{1762289}a^{9}+\frac{4370836}{1762289}a^{8}-\frac{49096}{19801}a^{7}-\frac{4370836}{1762289}a^{6}-\frac{1028432}{1762289}a^{5}+\frac{1028432}{1762289}a^{4}+\frac{1028432}{1762289}a^{3}+\frac{257108}{1762289}a^{2}-\frac{257108}{1762289}a+\frac{1505181}{1762289}$, $\frac{317811}{3524578}a^{39}+\frac{1597}{89}a^{28}+\frac{317811}{89}a^{17}-\frac{63244389}{3524578}a^{6}-1$, $\frac{28657}{3524578}a^{34}+\frac{144}{89}a^{23}+\frac{28657}{89}a^{12}-\frac{5702743}{3524578}a+1$, $\frac{831985}{3524578}a^{39}+\frac{158933}{1762289}a^{38}-\frac{2178257}{3524578}a^{37}+\frac{1028348}{1762289}a^{36}+\frac{2057137}{3524578}a^{35}-\frac{4370836}{1762289}a^{34}+\frac{8758669}{3524578}a^{33}+\frac{4371414}{1762289}a^{32}-\frac{18515159}{1762289}a^{31}+\frac{18510263}{1762289}a^{30}+\frac{18515516}{1762289}a^{29}+\frac{4365366}{1762289}a^{28}+\frac{110039202}{1762289}a^{27}-\frac{138308268}{1762289}a^{26}-\frac{127542483}{1762289}a^{25}+\frac{536841419}{1762289}a^{24}-\frac{6040100}{19801}a^{23}-\frac{535664474}{1762289}a^{22}+\frac{2276948465}{1762289}a^{21}-\frac{2277178492}{1762289}a^{20}-\frac{2277249535}{1762289}a^{19}+\frac{9644892692}{1762289}a^{18}+\frac{6830703461}{1762289}a^{17}-\frac{3352360237}{1762289}a^{16}-\frac{2276177276}{1762289}a^{15}-\frac{126267092}{1762289}a^{14}-\frac{129599211}{1762289}a^{13}-\frac{306221}{19801}a^{12}+\frac{383354993}{1762289}a^{11}+\frac{18515737}{1762289}a^{10}-\frac{115442924}{1762289}a^{9}+\frac{4370836}{1762289}a^{8}-\frac{4370479}{1762289}a^{7}-\frac{8741231}{3524578}a^{6}+\frac{18515159}{1762289}a^{5}+\frac{26214513}{3524578}a^{4}-\frac{6436608}{1762289}a^{3}-\frac{8713249}{3524578}a^{2}-\frac{257159}{1762289}a-\frac{514195}{3524578}$, $\frac{13}{3524578}a^{39}+\frac{13}{3524578}a^{38}-\frac{26}{1762289}a^{37}+\frac{25}{1762289}a^{36}+\frac{26}{1762289}a^{35}-\frac{221}{3524578}a^{34}+\frac{221}{3524578}a^{33}+\frac{221}{3524578}a^{32}-\frac{468}{1762289}a^{31}+\frac{468}{1762289}a^{30}-\frac{1651}{3524578}a^{29}-\frac{689}{1762289}a^{28}+\frac{3276}{1762289}a^{27}-\frac{6383}{3524578}a^{26}-\frac{3224}{1762289}a^{25}+\frac{13572}{1762289}a^{24}-\frac{27183}{3524578}a^{23}-\frac{13585}{1762289}a^{22}+\frac{57564}{1762289}a^{21}-\frac{115115}{3524578}a^{20}-\frac{57564}{1762289}a^{19}-\frac{13585}{1762289}a^{18}+\frac{27183}{3524578}a^{17}+\frac{13572}{1762289}a^{16}+\frac{3224}{1762289}a^{15}-\frac{6383}{3524578}a^{14}-\frac{3276}{1762289}a^{13}-\frac{689}{1762289}a^{12}+\frac{1651}{3524578}a^{11}+\frac{468}{1762289}a^{10}+\frac{468}{1762289}a^{9}+\frac{221}{3524578}a^{8}-\frac{51167188}{1762289}a^{7}-\frac{221}{3524578}a^{6}-\frac{26}{1762289}a^{5}+\frac{26}{1762289}a^{4}+\frac{7465202}{1762289}a^{3}+\frac{13}{3524578}a^{2}-\frac{13}{3524578}a-\frac{13}{3524578}$, $\frac{1}{3524578}a^{35}-\frac{9227465}{3524578}a^{2}+1$, $\frac{17}{1762289}a^{39}-\frac{34}{1762289}a^{38}+\frac{60695}{1762289}a^{37}+\frac{85}{1762289}a^{36}-\frac{136}{1762289}a^{35}+\frac{199}{39602}a^{34}+\frac{357}{1762289}a^{33}-\frac{578}{1762289}a^{32}+\frac{3383}{1762289}a^{31}+\frac{17}{19801}a^{30}-\frac{2448}{1762289}a^{29}+\frac{3383}{1762289}a^{28}-\frac{357}{1762289}a^{27}+\frac{12068240}{1762289}a^{26}+\frac{16915}{1762289}a^{25}+\frac{85}{1762289}a^{24}+\frac{1718361}{1762289}a^{23}+\frac{71043}{1762289}a^{22}-\frac{17}{1762289}a^{21}+\frac{487152}{1762289}a^{20}+\frac{3383}{19801}a^{19}+\frac{17}{1762289}a^{18}-\frac{115022}{1762289}a^{17}-\frac{71077}{1762289}a^{16}+\frac{2403702844}{1762289}a^{15}+\frac{27064}{1762289}a^{14}+\frac{16779}{1762289}a^{13}+\frac{350695732}{1762289}a^{12}-\frac{6766}{1762289}a^{11}-\frac{3961}{1762289}a^{10}+\frac{133954700}{1762289}a^{9}+\frac{935}{1762289}a^{7}+\frac{578}{1762289}a^{6}+\frac{84}{1762289}a^{4}-\frac{136}{1762289}a^{3}+\frac{191}{3524578}a+\frac{34}{1762289}$, $\frac{832019}{3524578}a^{39}+\frac{832019}{3524578}a^{38}-\frac{1664038}{1762289}a^{37}+\frac{1664039}{1762289}a^{36}+\frac{1664038}{1762289}a^{35}-\frac{14144323}{3524578}a^{34}+\frac{14144323}{3524578}a^{33}+\frac{14144323}{3524578}a^{32}-\frac{29952684}{1762289}a^{31}+\frac{29952684}{1762289}a^{30}+\frac{673141}{39602}a^{29}-\frac{44097007}{1762289}a^{28}+\frac{209668788}{1762289}a^{27}-\frac{408521329}{3524578}a^{26}-\frac{206340712}{1762289}a^{25}+\frac{868627836}{1762289}a^{24}-\frac{1739751729}{3524578}a^{23}-\frac{869459855}{1762289}a^{22}+\frac{3684180132}{1762289}a^{21}-\frac{7367528245}{3524578}a^{20}-\frac{3684180132}{1762289}a^{19}+\frac{175345665}{19801}a^{18}+\frac{1739751729}{3524578}a^{17}+\frac{868627836}{1762289}a^{16}+\frac{206340712}{1762289}a^{15}-\frac{408521329}{3524578}a^{14}-\frac{209668788}{1762289}a^{13}-\frac{44097007}{1762289}a^{12}+\frac{105666413}{3524578}a^{11}+\frac{29952684}{1762289}a^{10}+\frac{29952684}{1762289}a^{9}+\frac{14144323}{3524578}a^{8}-\frac{79439}{19801}a^{7}-\frac{14144323}{3524578}a^{6}-\frac{1664038}{1762289}a^{5}+\frac{1664038}{1762289}a^{4}-\frac{5801138}{1762289}a^{3}+\frac{832019}{3524578}a^{2}-\frac{832019}{3524578}a+\frac{2692559}{3524578}$, $\frac{17}{1762289}a^{39}-\frac{34}{1762289}a^{38}+\frac{85}{1762289}a^{36}-\frac{23320}{1762289}a^{35}+\frac{715}{3524578}a^{33}-\frac{578}{1762289}a^{32}+\frac{3383}{1762289}a^{31}+\frac{17}{19801}a^{30}-\frac{2448}{1762289}a^{29}+\frac{3383}{1762289}a^{28}-\frac{357}{1762289}a^{27}-\frac{10370}{1762289}a^{26}+\frac{16915}{1762289}a^{25}-\frac{4613548}{1762289}a^{24}-\frac{43928}{1762289}a^{23}+\frac{71043}{1762289}a^{22}-\frac{17}{1762289}a^{21}+\frac{487152}{1762289}a^{20}+\frac{3383}{19801}a^{19}+\frac{17}{1762289}a^{18}-\frac{115022}{1762289}a^{17}-\frac{71077}{1762289}a^{16}+\frac{51}{1762289}a^{15}+\frac{27064}{1762289}a^{14}-\frac{918115989}{1762289}a^{13}+\frac{221}{1762289}a^{12}-\frac{6766}{1762289}a^{11}-\frac{3961}{1762289}a^{10}+\frac{133954700}{1762289}a^{9}+\frac{935}{1762289}a^{7}+\frac{578}{1762289}a^{6}-\frac{221}{1762289}a^{4}-\frac{136}{1762289}a^{3}+\frac{4613616}{1762289}a^{2}+\frac{51}{1762289}a-\frac{5702819}{3524578}$, $\frac{4}{1762289}a^{39}+\frac{98209}{1762289}a^{38}-\frac{1}{3524578}a^{33}+\frac{987}{89}a^{27}+\frac{196418}{89}a^{16}-\frac{31622993}{1762289}a^{6}-\frac{19543591}{1762289}a^{5}+\frac{2178309}{3524578}$, $\frac{832019}{3524578}a^{39}-\frac{832019}{3524578}a^{38}-\frac{3}{3524578}a^{37}+\frac{1248028}{1762289}a^{36}-\frac{4160095}{3524578}a^{35}+\frac{10816247}{3524578}a^{33}-\frac{17472399}{3524578}a^{32}+\frac{4181}{3524578}a^{31}+\frac{22882613}{1762289}a^{30}-\frac{832019}{39602}a^{29}+\frac{165571781}{3524578}a^{28}+\frac{14144323}{1762289}a^{27}-\frac{313671163}{3524578}a^{26}+\frac{496715343}{3524578}a^{25}-\frac{3328076}{1762289}a^{24}-\frac{1328734343}{3524578}a^{23}+\frac{2152433153}{3524578}a^{22}+\frac{832019}{1762289}a^{21}-\frac{5627776893}{3524578}a^{20}+\frac{9107279597}{3524578}a^{19}+\frac{9106447955}{3524578}a^{17}+\frac{5628608535}{3524578}a^{16}+\frac{832019}{1762289}a^{15}-\frac{2152433153}{3524578}a^{14}-\frac{1328734343}{3524578}a^{13}+\frac{3328076}{1762289}a^{12}+\frac{496715343}{3524578}a^{11}+\frac{313671163}{3524578}a^{10}+\frac{96932304}{1762289}a^{9}+\frac{4181}{3524578}a^{8}-\frac{832019}{39602}a^{7}-\frac{45761045}{3524578}a^{6}+\frac{20815108}{1762289}a^{4}+\frac{10021856}{1762289}a^{3}-\frac{4160095}{3524578}a-\frac{2496057}{3524578}$, $\frac{158933}{1762289}a^{39}-\frac{55}{1762289}a^{38}+\frac{3}{3524578}a^{37}+\frac{275}{3524578}a^{36}-\frac{220}{1762289}a^{35}-\frac{23184}{1762289}a^{34}+\frac{1155}{3524578}a^{33}-\frac{935}{1762289}a^{32}+\frac{10945}{3524578}a^{31}+\frac{55}{39602}a^{30}-\frac{3960}{1762289}a^{29}+\frac{63255339}{3524578}a^{28}-\frac{1155}{3524578}a^{27}-\frac{16775}{1762289}a^{26}+\frac{54725}{3524578}a^{25}+\frac{275}{3524578}a^{24}-\frac{52637}{19801}a^{23}+\frac{229845}{3524578}a^{22}-\frac{55}{3524578}a^{21}+\frac{788040}{1762289}a^{20}+\frac{10945}{39602}a^{19}+\frac{55}{3524578}a^{18}+\frac{6292789546}{1762289}a^{17}-\frac{229955}{3524578}a^{16}+\frac{165}{3524578}a^{15}+\frac{43780}{1762289}a^{14}+\frac{54285}{3524578}a^{13}-\frac{20632189}{39602}a^{12}-\frac{10945}{1762289}a^{11}-\frac{12815}{3524578}a^{10}+\frac{433486517}{3524578}a^{9}+\frac{3025}{3524578}a^{7}-\frac{63242519}{3524578}a^{6}-\frac{12079266}{1762289}a^{4}-\frac{220}{1762289}a^{3}+\frac{9227397}{3524578}a+\frac{55}{1762289}$, $\frac{75023}{3524578}a^{36}-\frac{14328}{1762289}a^{35}-\frac{199}{39602}a^{34}+\frac{199}{39602}a^{33}+\frac{377}{89}a^{25}-\frac{144}{89}a^{24}-a^{23}+a^{22}+\frac{75025}{89}a^{14}-\frac{28657}{89}a^{13}-199a^{12}+199a^{11}+\frac{377}{3524578}a^{3}-\frac{72}{1762289}a^{2}-\frac{1}{39602}a+\frac{1}{39602}$, $\frac{832011}{3524578}a^{39}-\frac{733810}{1762289}a^{38}+\frac{4160095}{3524578}a^{36}-\frac{3342404}{1762289}a^{35}+\frac{17472399}{3524578}a^{33}-\frac{14144323}{1762289}a^{32}-\frac{4181}{3524578}a^{31}+\frac{832019}{39602}a^{30}-\frac{59905368}{1762289}a^{29}+\frac{165571781}{3524578}a^{28}+\frac{21614775}{3524578}a^{27}-\frac{253765795}{1762289}a^{26}+\frac{827858905}{3524578}a^{25}-\frac{1542593}{3524578}a^{24}-\frac{1074968548}{1762289}a^{23}+\frac{3477007401}{3524578}a^{22}-\frac{832019}{3524578}a^{21}-\frac{4554055808}{1762289}a^{20}+\frac{165571781}{39602}a^{19}+\frac{832019}{3524578}a^{18}-\frac{2814720277}{1762289}a^{17}+\frac{4299874197}{3524578}a^{16}+\frac{2496057}{3524578}a^{15}+\frac{662287124}{1762289}a^{14}-\frac{313671761}{3524578}a^{13}+\frac{10816247}{3524578}a^{12}-\frac{165571781}{1762289}a^{11}-\frac{193860427}{3524578}a^{10}-\frac{119814917}{3524578}a^{9}+\frac{45761045}{3524578}a^{7}+\frac{45767316}{1762289}a^{6}-\frac{19543591}{1762289}a^{5}-\frac{10816247}{3524578}a^{4}-\frac{3328076}{1762289}a^{3}-\frac{72}{1762289}a^{2}+\frac{2496057}{3524578}a+\frac{832019}{1762289}$, $\frac{832027}{3524578}a^{39}-\frac{930224}{1762289}a^{38}-\frac{16}{1762289}a^{37}+\frac{46743}{39602}a^{36}-\frac{3328060}{1762289}a^{35}-\frac{68}{1762289}a^{34}+\frac{17472535}{3524578}a^{33}-\frac{14144255}{1762289}a^{32}-\frac{4757}{3524578}a^{31}+\frac{74050267}{3524578}a^{30}-\frac{59905876}{1762289}a^{29}+\frac{165570933}{3524578}a^{28}-\frac{56555541}{3524578}a^{27}-\frac{253767759}{1762289}a^{26}+\frac{827854937}{3524578}a^{25}+\frac{4176799}{3524578}a^{24}-\frac{1074976912}{1762289}a^{23}+\frac{3476990681}{3524578}a^{22}-\frac{761171}{3524578}a^{21}-\frac{4554091228}{1762289}a^{20}+\frac{14735817661}{3524578}a^{19}+\frac{815299}{3524578}a^{18}-\frac{2814711913}{1762289}a^{17}-\frac{11257200371}{3524578}a^{16}+\frac{2500025}{3524578}a^{15}+\frac{662285160}{1762289}a^{14}+\frac{821198721}{3524578}a^{13}+\frac{10815399}{3524578}a^{12}-\frac{165571273}{1762289}a^{11}-\frac{193859851}{3524578}a^{10}-\frac{119814341}{3524578}a^{9}+\frac{68}{1762289}a^{8}-\frac{17485077}{3524578}a^{7}+\frac{14144255}{1762289}a^{6}+\frac{19543575}{1762289}a^{5}-\frac{10816215}{3524578}a^{4}-\frac{3328060}{1762289}a^{3}+\frac{4}{1762289}a^{2}+\frac{2496049}{3524578}a-\frac{930274}{1762289}$ 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:  \( 138297149643015.1 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 138297149643015.1 \cdot 22}{66\cdot\sqrt{10279259898673041257519092860384627329993230987644195556640625}}\cr\approx \mathstrut & 0.132223429782772 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + 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^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + 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^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + 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^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + 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

$C_2^2\times C_{10}$ (as 40T7):

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)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.741200625.1, 10.10.1790566527853125.1, 10.10.669871503125.1, \(\Q(\zeta_{33})^+\), 10.0.162778775259375.1, \(\Q(\zeta_{11})\), 10.0.52089208083.1, 10.0.7368586534375.1, 20.20.3206128490667995866421572265625.1, 20.0.3206128490667995866421572265625.3, 20.0.3206128490667995866421572265625.2, 20.0.26496929674942114598525390625.1, 20.0.54296067514572573056640625.1, 20.0.3206128490667995866421572265625.1, \(\Q(\zeta_{33})\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{4}$ R R ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{8}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(5\) Copy content Toggle raw display Deg $20$$2$$10$$10$
Deg $20$$2$$10$$10$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$