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Properties

Label 44.0.169...853.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.695\times 10^{91}$
Root discriminant \(118.41\)
Ramified primes $13,23$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647)
 
gp: K = bnfinit(y^44 - 7*y^43 + 158*y^41 - 411*y^40 - 924*y^39 + 5782*y^38 - 3076*y^37 - 24525*y^36 + 51385*y^35 - 5991*y^34 + 22269*y^33 + 236728*y^32 - 1028729*y^31 + 2848698*y^30 + 2635339*y^29 - 5513154*y^28 + 30800385*y^27 + 1371999*y^26 + 59487017*y^25 + 257798043*y^24 - 27985708*y^23 + 932537292*y^22 + 1017917378*y^21 + 271225362*y^20 + 5026501765*y^19 - 1526792896*y^18 + 643104807*y^17 + 9058106871*y^16 - 22148706708*y^15 + 31881217675*y^14 + 37375872227*y^13 + 2814647127*y^12 + 110216158894*y^11 - 112725908932*y^10 - 222555468015*y^9 + 365357483551*y^8 + 138214465517*y^7 - 150778794282*y^6 - 65026060302*y^5 + 169684727196*y^4 - 41892262778*y^3 + 67861092927*y^2 - 19812374035*y + 27694856647, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647)
 

\( x^{44} - 7 x^{43} + 158 x^{41} - 411 x^{40} - 924 x^{39} + 5782 x^{38} - 3076 x^{37} + \cdots + 27694856647 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(169\!\cdots\!853\) \(\medspace = 13^{33}\cdot 23^{40}\) 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:  \(118.41\)
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:  $13^{3/4}23^{10/11}\approx 118.410962317434$
Ramified primes:   \(13\), \(23\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(1,·)$, $\chi_{299}(259,·)$, $\chi_{299}(261,·)$, $\chi_{299}(8,·)$, $\chi_{299}(265,·)$, $\chi_{299}(12,·)$, $\chi_{299}(142,·)$, $\chi_{299}(144,·)$, $\chi_{299}(18,·)$, $\chi_{299}(131,·)$, $\chi_{299}(278,·)$, $\chi_{299}(151,·)$, $\chi_{299}(25,·)$, $\chi_{299}(27,·)$, $\chi_{299}(285,·)$, $\chi_{299}(31,·)$, $\chi_{299}(164,·)$, $\chi_{299}(294,·)$, $\chi_{299}(170,·)$, $\chi_{299}(174,·)$, $\chi_{299}(47,·)$, $\chi_{299}(177,·)$, $\chi_{299}(187,·)$, $\chi_{299}(190,·)$, $\chi_{299}(64,·)$, $\chi_{299}(196,·)$, $\chi_{299}(118,·)$, $\chi_{299}(70,·)$, $\chi_{299}(200,·)$, $\chi_{299}(73,·)$, $\chi_{299}(77,·)$, $\chi_{299}(209,·)$, $\chi_{299}(213,·)$, $\chi_{299}(216,·)$, $\chi_{299}(220,·)$, $\chi_{299}(96,·)$, $\chi_{299}(105,·)$, $\chi_{299}(239,·)$, $\chi_{299}(242,·)$, $\chi_{299}(116,·)$, $\chi_{299}(246,·)$, $\chi_{299}(233,·)$, $\chi_{299}(248,·)$, $\chi_{299}(255,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{3}a^{33}+\frac{1}{3}a^{31}+\frac{1}{3}a^{29}+\frac{1}{3}a^{27}+\frac{1}{3}a^{21}-\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{34}+\frac{1}{3}a^{32}+\frac{1}{3}a^{30}+\frac{1}{3}a^{28}+\frac{1}{3}a^{22}-\frac{1}{3}a^{20}-\frac{1}{3}a^{16}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{35}-\frac{1}{3}a^{27}+\frac{1}{3}a^{23}+\frac{1}{3}a^{21}+\frac{1}{3}a^{19}-\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{36}-\frac{1}{3}a^{28}+\frac{1}{3}a^{24}+\frac{1}{3}a^{22}+\frac{1}{3}a^{20}-\frac{1}{3}a^{18}+\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{37}-\frac{1}{3}a^{29}+\frac{1}{3}a^{25}+\frac{1}{3}a^{23}+\frac{1}{3}a^{21}-\frac{1}{3}a^{19}+\frac{1}{3}a^{17}+\frac{1}{3}a^{15}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{38}-\frac{1}{3}a^{30}+\frac{1}{3}a^{26}+\frac{1}{3}a^{24}+\frac{1}{3}a^{22}-\frac{1}{3}a^{20}+\frac{1}{3}a^{18}+\frac{1}{3}a^{16}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{39}-\frac{1}{3}a^{31}+\frac{1}{3}a^{27}+\frac{1}{3}a^{25}+\frac{1}{3}a^{23}-\frac{1}{3}a^{21}+\frac{1}{3}a^{19}+\frac{1}{3}a^{17}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{417}a^{40}-\frac{22}{139}a^{39}+\frac{16}{417}a^{38}-\frac{8}{139}a^{37}+\frac{41}{417}a^{36}-\frac{55}{417}a^{35}-\frac{5}{417}a^{34}-\frac{11}{139}a^{33}-\frac{49}{139}a^{32}+\frac{51}{139}a^{31}-\frac{52}{139}a^{30}+\frac{37}{139}a^{29}+\frac{3}{139}a^{28}-\frac{107}{417}a^{27}-\frac{10}{417}a^{26}+\frac{42}{139}a^{25}+\frac{94}{417}a^{24}-\frac{1}{3}a^{23}+\frac{13}{139}a^{22}+\frac{107}{417}a^{21}-\frac{182}{417}a^{20}+\frac{146}{417}a^{19}-\frac{3}{139}a^{18}-\frac{200}{417}a^{17}+\frac{68}{417}a^{16}+\frac{143}{417}a^{15}+\frac{97}{417}a^{14}-\frac{94}{417}a^{13}-\frac{185}{417}a^{12}-\frac{5}{139}a^{11}+\frac{37}{139}a^{10}-\frac{206}{417}a^{9}+\frac{27}{139}a^{8}+\frac{94}{417}a^{7}-\frac{24}{139}a^{6}-\frac{19}{139}a^{5}-\frac{167}{417}a^{4}-\frac{194}{417}a^{3}-\frac{148}{417}a^{2}+\frac{55}{139}a-\frac{1}{3}$, $\frac{1}{417}a^{41}-\frac{31}{417}a^{39}+\frac{59}{417}a^{38}-\frac{14}{417}a^{37}+\frac{10}{417}a^{36}-\frac{7}{139}a^{35}+\frac{18}{139}a^{34}+\frac{38}{417}a^{33}+\frac{14}{139}a^{32}+\frac{73}{417}a^{31}-\frac{38}{417}a^{30}-\frac{57}{139}a^{29}-\frac{208}{417}a^{28}+\frac{52}{139}a^{27}+\frac{161}{417}a^{26}+\frac{70}{417}a^{25}-\frac{17}{139}a^{24}-\frac{100}{417}a^{23}-\frac{33}{139}a^{22}+\frac{23}{139}a^{21}-\frac{190}{417}a^{20}-\frac{103}{417}a^{19}+\frac{40}{417}a^{18}-\frac{22}{139}a^{17}+\frac{61}{139}a^{16}-\frac{65}{139}a^{15}-\frac{86}{417}a^{14}+\frac{5}{417}a^{13}+\frac{7}{417}a^{12}+\frac{94}{417}a^{11}+\frac{31}{417}a^{10}+\frac{107}{417}a^{9}-\frac{40}{139}a^{8}-\frac{41}{139}a^{7}-\frac{83}{417}a^{6}-\frac{176}{417}a^{5}+\frac{43}{417}a^{4}-\frac{25}{417}a^{3}-\frac{151}{417}a^{2}+\frac{187}{417}a$, $\frac{1}{249783}a^{42}+\frac{286}{249783}a^{41}+\frac{269}{249783}a^{40}-\frac{668}{249783}a^{39}+\frac{1922}{249783}a^{38}-\frac{10499}{249783}a^{37}-\frac{22669}{249783}a^{36}-\frac{31904}{249783}a^{35}+\frac{1927}{83261}a^{34}-\frac{3716}{249783}a^{33}+\frac{6488}{249783}a^{32}-\frac{114794}{249783}a^{31}+\frac{14590}{83261}a^{30}-\frac{2470}{249783}a^{29}-\frac{96247}{249783}a^{28}+\frac{33462}{83261}a^{27}+\frac{44228}{249783}a^{26}-\frac{9424}{83261}a^{25}+\frac{115679}{249783}a^{24}-\frac{95836}{249783}a^{23}-\frac{42260}{249783}a^{22}+\frac{19813}{249783}a^{21}+\frac{1601}{249783}a^{20}-\frac{23006}{83261}a^{19}+\frac{68027}{249783}a^{18}+\frac{115351}{249783}a^{17}+\frac{74072}{249783}a^{16}+\frac{65579}{249783}a^{15}+\frac{93052}{249783}a^{14}-\frac{1604}{249783}a^{13}-\frac{113591}{249783}a^{12}+\frac{30755}{249783}a^{11}-\frac{45019}{249783}a^{10}+\frac{120470}{249783}a^{9}-\frac{99659}{249783}a^{8}-\frac{39865}{249783}a^{7}+\frac{107942}{249783}a^{6}+\frac{88843}{249783}a^{5}+\frac{97142}{249783}a^{4}+\frac{53761}{249783}a^{3}+\frac{67447}{249783}a^{2}-\frac{3666}{83261}a+\frac{1}{3}$, $\frac{1}{16\!\cdots\!29}a^{43}-\frac{44\!\cdots\!88}{55\!\cdots\!43}a^{42}+\frac{12\!\cdots\!21}{16\!\cdots\!29}a^{41}-\frac{56\!\cdots\!45}{55\!\cdots\!43}a^{40}-\frac{24\!\cdots\!28}{16\!\cdots\!29}a^{39}+\frac{63\!\cdots\!20}{55\!\cdots\!43}a^{38}-\frac{44\!\cdots\!92}{55\!\cdots\!43}a^{37}-\frac{97\!\cdots\!58}{16\!\cdots\!29}a^{36}+\frac{10\!\cdots\!58}{16\!\cdots\!29}a^{35}-\frac{23\!\cdots\!37}{16\!\cdots\!29}a^{34}-\frac{68\!\cdots\!02}{55\!\cdots\!43}a^{33}-\frac{25\!\cdots\!24}{16\!\cdots\!29}a^{32}-\frac{41\!\cdots\!66}{16\!\cdots\!29}a^{31}+\frac{16\!\cdots\!64}{16\!\cdots\!29}a^{30}+\frac{29\!\cdots\!14}{16\!\cdots\!29}a^{29}+\frac{60\!\cdots\!70}{55\!\cdots\!43}a^{28}-\frac{36\!\cdots\!35}{55\!\cdots\!43}a^{27}-\frac{16\!\cdots\!76}{55\!\cdots\!43}a^{26}+\frac{23\!\cdots\!10}{16\!\cdots\!29}a^{25}+\frac{58\!\cdots\!56}{16\!\cdots\!29}a^{24}+\frac{25\!\cdots\!86}{55\!\cdots\!43}a^{23}-\frac{88\!\cdots\!87}{16\!\cdots\!29}a^{22}-\frac{90\!\cdots\!87}{16\!\cdots\!29}a^{21}-\frac{25\!\cdots\!55}{55\!\cdots\!43}a^{20}-\frac{41\!\cdots\!19}{16\!\cdots\!29}a^{19}+\frac{28\!\cdots\!81}{16\!\cdots\!29}a^{18}-\frac{26\!\cdots\!94}{55\!\cdots\!43}a^{17}+\frac{19\!\cdots\!35}{55\!\cdots\!43}a^{16}-\frac{24\!\cdots\!87}{55\!\cdots\!43}a^{15}+\frac{52\!\cdots\!81}{16\!\cdots\!29}a^{14}-\frac{81\!\cdots\!25}{16\!\cdots\!29}a^{13}-\frac{55\!\cdots\!73}{16\!\cdots\!29}a^{12}-\frac{97\!\cdots\!69}{16\!\cdots\!29}a^{11}+\frac{66\!\cdots\!05}{16\!\cdots\!29}a^{10}+\frac{88\!\cdots\!32}{16\!\cdots\!29}a^{9}-\frac{58\!\cdots\!96}{16\!\cdots\!29}a^{8}-\frac{18\!\cdots\!77}{55\!\cdots\!43}a^{7}+\frac{21\!\cdots\!79}{55\!\cdots\!43}a^{6}+\frac{19\!\cdots\!58}{55\!\cdots\!43}a^{5}+\frac{74\!\cdots\!96}{16\!\cdots\!29}a^{4}-\frac{17\!\cdots\!72}{55\!\cdots\!43}a^{3}+\frac{19\!\cdots\!19}{55\!\cdots\!43}a^{2}+\frac{59\!\cdots\!61}{16\!\cdots\!29}a-\frac{33\!\cdots\!13}{82\!\cdots\!73}$ 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

not computed

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:  $21$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647)
 
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^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647, 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^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647);
 
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^44 - 7*x^43 + 158*x^41 - 411*x^40 - 924*x^39 + 5782*x^38 - 3076*x^37 - 24525*x^36 + 51385*x^35 - 5991*x^34 + 22269*x^33 + 236728*x^32 - 1028729*x^31 + 2848698*x^30 + 2635339*x^29 - 5513154*x^28 + 30800385*x^27 + 1371999*x^26 + 59487017*x^25 + 257798043*x^24 - 27985708*x^23 + 932537292*x^22 + 1017917378*x^21 + 271225362*x^20 + 5026501765*x^19 - 1526792896*x^18 + 643104807*x^17 + 9058106871*x^16 - 22148706708*x^15 + 31881217675*x^14 + 37375872227*x^13 + 2814647127*x^12 + 110216158894*x^11 - 112725908932*x^10 - 222555468015*x^9 + 365357483551*x^8 + 138214465517*x^7 - 150778794282*x^6 - 65026060302*x^5 + 169684727196*x^4 - 41892262778*x^3 + 67861092927*x^2 - 19812374035*x + 27694856647);
 
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

$C_{44}$ (as 44T1):

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 cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, \(\Q(\zeta_{23})^+\), 22.22.3075626510913487571920886830127053316437.1

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 $44$ ${\href{/padicField/3.11.0.1}{11} }^{4}$ $44$ $44$ $44$ R $22^{2}$ $44$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ $44$ $44$ $44$ $22^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ $44$

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
\(13\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(23\) Copy content Toggle raw display 23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$
23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$