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Properties

Label 38.38.261...857.1
Degree $38$
Signature $[38, 0]$
Discriminant $2.612\times 10^{98}$
Root discriminant \(388.97\)
Ramified prime $457$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127)
 
gp: K = bnfinit(y^38 - y^37 - 222*y^36 + 479*y^35 + 21047*y^34 - 66053*y^33 - 1110228*y^32 + 4495506*y^31 + 35823405*y^30 - 180946047*y^29 - 731415663*y^28 + 4712734359*y^27 + 9238434306*y^26 - 83665939703*y^25 - 59973164681*y^24 + 1044644450973*y^23 - 101098208429*y^22 - 9336414182461*y^21 + 6288159462034*y^20 + 60150648390199*y^19 - 67252981484387*y^18 - 278611442416743*y^17 + 417500804951451*y^16 + 915079337067075*y^15 - 1716002791423036*y^14 - 2065408719688167*y^13 + 4825202788269126*y^12 + 2982003177766905*y^11 - 9269437748788275*y^10 - 2187058821094830*y^9 + 11855288436035595*y^8 - 411608485430453*y^7 - 9515695267587717*y^6 + 2357948299236310*y^5 + 4196479083397002*y^4 - 1892165812751211*y^3 - 650631784472963*y^2 + 526610168112801*y - 83497743723127, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127)
 

\( x^{38} - x^{37} - 222 x^{36} + 479 x^{35} + 21047 x^{34} - 66053 x^{33} - 1110228 x^{32} + \cdots - 83497743723127 \) Copy content Toggle raw display

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

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[38, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(261\!\cdots\!857\) \(\medspace = 457^{37}\) 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:  \(388.97\)
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:  $457^{37/38}\approx 388.9720585765065$
Ramified primes:   \(457\) 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:  \(\Q(\sqrt{457}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(457\)
Dirichlet character group:    $\lbrace$$\chi_{457}(256,·)$, $\chi_{457}(1,·)$, $\chi_{457}(4,·)$, $\chi_{457}(389,·)$, $\chi_{457}(257,·)$, $\chi_{457}(393,·)$, $\chi_{457}(215,·)$, $\chi_{457}(16,·)$, $\chi_{457}(17,·)$, $\chi_{457}(403,·)$, $\chi_{457}(407,·)$, $\chi_{457}(283,·)$, $\chi_{457}(415,·)$, $\chi_{457}(289,·)$, $\chi_{457}(168,·)$, $\chi_{457}(42,·)$, $\chi_{457}(114,·)$, $\chi_{457}(174,·)$, $\chi_{457}(200,·)$, $\chi_{457}(50,·)$, $\chi_{457}(54,·)$, $\chi_{457}(440,·)$, $\chi_{457}(185,·)$, $\chi_{457}(64,·)$, $\chi_{457}(441,·)$, $\chi_{457}(68,·)$, $\chi_{457}(453,·)$, $\chi_{457}(456,·)$, $\chi_{457}(201,·)$, $\chi_{457}(343,·)$, $\chi_{457}(216,·)$, $\chi_{457}(218,·)$, $\chi_{457}(347,·)$, $\chi_{457}(272,·)$, $\chi_{457}(110,·)$, $\chi_{457}(239,·)$, $\chi_{457}(241,·)$, $\chi_{457}(242,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{109}a^{31}-\frac{5}{109}a^{30}-\frac{30}{109}a^{29}+\frac{27}{109}a^{28}+\frac{43}{109}a^{27}+\frac{47}{109}a^{26}-\frac{44}{109}a^{25}-\frac{14}{109}a^{24}+\frac{4}{109}a^{23}+\frac{50}{109}a^{22}+\frac{35}{109}a^{21}-\frac{15}{109}a^{20}+\frac{34}{109}a^{19}-\frac{28}{109}a^{18}+\frac{6}{109}a^{17}+\frac{16}{109}a^{16}-\frac{31}{109}a^{15}+\frac{45}{109}a^{14}+\frac{36}{109}a^{13}-\frac{14}{109}a^{12}+\frac{19}{109}a^{11}-\frac{25}{109}a^{10}-\frac{52}{109}a^{9}-\frac{18}{109}a^{8}+\frac{23}{109}a^{7}+\frac{11}{109}a^{6}-\frac{10}{109}a^{5}-\frac{11}{109}a^{4}-\frac{2}{109}a^{3}+\frac{25}{109}a^{2}+\frac{6}{109}a+\frac{8}{109}$, $\frac{1}{109}a^{32}+\frac{54}{109}a^{30}-\frac{14}{109}a^{29}-\frac{40}{109}a^{28}+\frac{44}{109}a^{27}-\frac{27}{109}a^{26}-\frac{16}{109}a^{25}+\frac{43}{109}a^{24}-\frac{39}{109}a^{23}-\frac{42}{109}a^{22}+\frac{51}{109}a^{21}-\frac{41}{109}a^{20}+\frac{33}{109}a^{19}-\frac{25}{109}a^{18}+\frac{46}{109}a^{17}+\frac{49}{109}a^{16}-\frac{1}{109}a^{15}+\frac{43}{109}a^{14}-\frac{52}{109}a^{13}-\frac{51}{109}a^{12}-\frac{39}{109}a^{11}+\frac{41}{109}a^{10}+\frac{49}{109}a^{9}+\frac{42}{109}a^{8}+\frac{17}{109}a^{7}+\frac{45}{109}a^{6}+\frac{48}{109}a^{5}+\frac{52}{109}a^{4}+\frac{15}{109}a^{3}+\frac{22}{109}a^{2}+\frac{38}{109}a+\frac{40}{109}$, $\frac{1}{13843}a^{33}-\frac{58}{13843}a^{32}+\frac{14}{13843}a^{31}+\frac{3812}{13843}a^{30}-\frac{535}{13843}a^{29}+\frac{3464}{13843}a^{28}+\frac{5729}{13843}a^{27}+\frac{433}{13843}a^{26}-\frac{2719}{13843}a^{25}+\frac{2605}{13843}a^{24}-\frac{4044}{13843}a^{23}+\frac{4520}{13843}a^{22}-\frac{3091}{13843}a^{21}-\frac{2548}{13843}a^{20}+\frac{298}{13843}a^{19}-\frac{8}{127}a^{18}+\frac{520}{13843}a^{17}-\frac{976}{13843}a^{16}+\frac{5810}{13843}a^{15}-\frac{1294}{13843}a^{14}-\frac{1}{13843}a^{13}+\frac{5114}{13843}a^{12}-\frac{419}{13843}a^{11}-\frac{3945}{13843}a^{10}-\frac{1483}{13843}a^{9}+\frac{1244}{13843}a^{8}+\frac{428}{13843}a^{7}-\frac{2130}{13843}a^{6}+\frac{3118}{13843}a^{5}-\frac{6158}{13843}a^{4}-\frac{6218}{13843}a^{3}+\frac{923}{13843}a^{2}+\frac{4681}{13843}a+\frac{35}{109}$, $\frac{1}{13843}a^{34}-\frac{48}{13843}a^{32}+\frac{52}{13843}a^{31}+\frac{6439}{13843}a^{30}-\frac{5849}{13843}a^{29}+\frac{6489}{13843}a^{28}+\frac{4547}{13843}a^{27}-\frac{4783}{13843}a^{26}-\frac{6761}{13843}a^{25}-\frac{6878}{13843}a^{24}-\frac{3337}{13843}a^{23}+\frac{2529}{13843}a^{22}+\frac{6515}{13843}a^{21}-\frac{6643}{13843}a^{20}-\frac{2384}{13843}a^{19}-\frac{4590}{13843}a^{18}+\frac{1371}{13843}a^{17}-\frac{3681}{13843}a^{16}+\frac{3454}{13843}a^{15}-\frac{377}{13843}a^{14}+\frac{992}{13843}a^{13}-\frac{2003}{13843}a^{12}+\frac{5281}{13843}a^{11}+\frac{5546}{13843}a^{10}-\frac{3617}{13843}a^{9}+\frac{2857}{13843}a^{8}+\frac{1358}{13843}a^{7}+\frac{5562}{13843}a^{6}+\frac{5141}{13843}a^{5}-\frac{2956}{13843}a^{4}+\frac{3499}{13843}a^{3}+\frac{2716}{13843}a^{2}+\frac{226}{13843}a-\frac{52}{109}$, $\frac{1}{13843}a^{35}+\frac{62}{13843}a^{32}-\frac{1}{13843}a^{31}+\frac{3645}{13843}a^{30}+\frac{2780}{13843}a^{29}+\frac{5465}{13843}a^{28}+\frac{4271}{13843}a^{27}+\frac{5768}{13843}a^{26}+\frac{6247}{13843}a^{25}-\frac{4662}{13843}a^{24}+\frac{3235}{13843}a^{23}-\frac{299}{13843}a^{22}+\frac{1580}{13843}a^{21}+\frac{5868}{13843}a^{20}-\frac{1462}{13843}a^{19}+\frac{5743}{13843}a^{18}-\frac{3613}{13843}a^{17}-\frac{6437}{13843}a^{16}-\frac{2167}{13843}a^{15}+\frac{1999}{13843}a^{14}-\frac{1924}{13843}a^{13}+\frac{182}{13843}a^{12}+\frac{4357}{13843}a^{11}+\frac{2476}{13843}a^{10}-\frac{4573}{13843}a^{9}+\frac{1888}{13843}a^{8}-\frac{6914}{13843}a^{7}+\frac{5771}{13843}a^{6}+\frac{5865}{13843}a^{5}+\frac{650}{13843}a^{4}-\frac{4283}{13843}a^{3}-\frac{2587}{13843}a^{2}+\frac{4724}{13843}a+\frac{41}{109}$, $\frac{1}{13843}a^{36}+\frac{39}{13843}a^{32}-\frac{17}{13843}a^{31}+\frac{3672}{13843}a^{30}+\frac{6123}{13843}a^{29}-\frac{5265}{13843}a^{28}-\frac{3101}{13843}a^{27}-\frac{533}{13843}a^{26}-\frac{2327}{13843}a^{25}+\frac{4793}{13843}a^{24}+\frac{4176}{13843}a^{23}-\frac{5991}{13843}a^{22}+\frac{1422}{13843}a^{21}-\frac{1855}{13843}a^{20}-\frac{3589}{13843}a^{19}-\frac{3905}{13843}a^{18}+\frac{2471}{13843}a^{17}+\frac{5513}{13843}a^{16}-\frac{5034}{13843}a^{15}-\frac{6532}{13843}a^{14}+\frac{1514}{13843}a^{13}+\frac{4662}{13843}a^{12}+\frac{3308}{13843}a^{11}-\frac{2045}{13843}a^{10}-\frac{4337}{13843}a^{9}-\frac{3143}{13843}a^{8}+\frac{6794}{13843}a^{7}+\frac{2543}{13843}a^{6}-\frac{3182}{13843}a^{5}+\frac{1847}{13843}a^{4}+\frac{2945}{13843}a^{3}-\frac{6782}{13843}a^{2}+\frac{6069}{13843}a+\frac{22}{109}$, $\frac{1}{50\!\cdots\!13}a^{37}+\frac{34\!\cdots\!36}{50\!\cdots\!13}a^{36}-\frac{15\!\cdots\!05}{50\!\cdots\!13}a^{35}+\frac{61\!\cdots\!99}{50\!\cdots\!13}a^{34}-\frac{17\!\cdots\!45}{50\!\cdots\!13}a^{33}+\frac{22\!\cdots\!14}{50\!\cdots\!13}a^{32}-\frac{30\!\cdots\!68}{50\!\cdots\!13}a^{31}+\frac{19\!\cdots\!67}{50\!\cdots\!13}a^{30}+\frac{15\!\cdots\!10}{50\!\cdots\!13}a^{29}-\frac{11\!\cdots\!84}{50\!\cdots\!13}a^{28}+\frac{14\!\cdots\!56}{50\!\cdots\!13}a^{27}-\frac{24\!\cdots\!89}{50\!\cdots\!13}a^{26}+\frac{20\!\cdots\!72}{50\!\cdots\!13}a^{25}+\frac{11\!\cdots\!16}{50\!\cdots\!13}a^{24}+\frac{43\!\cdots\!88}{50\!\cdots\!13}a^{23}-\frac{24\!\cdots\!31}{50\!\cdots\!13}a^{22}+\frac{24\!\cdots\!32}{50\!\cdots\!13}a^{21}-\frac{23\!\cdots\!70}{50\!\cdots\!13}a^{20}-\frac{97\!\cdots\!60}{50\!\cdots\!13}a^{19}+\frac{31\!\cdots\!90}{50\!\cdots\!13}a^{18}+\frac{81\!\cdots\!20}{50\!\cdots\!13}a^{17}+\frac{21\!\cdots\!28}{50\!\cdots\!13}a^{16}+\frac{21\!\cdots\!12}{50\!\cdots\!13}a^{15}+\frac{65\!\cdots\!47}{50\!\cdots\!13}a^{14}+\frac{12\!\cdots\!82}{50\!\cdots\!13}a^{13}+\frac{22\!\cdots\!13}{50\!\cdots\!13}a^{12}-\frac{13\!\cdots\!75}{50\!\cdots\!13}a^{11}+\frac{13\!\cdots\!67}{50\!\cdots\!13}a^{10}+\frac{22\!\cdots\!94}{50\!\cdots\!13}a^{9}-\frac{23\!\cdots\!85}{50\!\cdots\!13}a^{8}-\frac{13\!\cdots\!81}{50\!\cdots\!13}a^{7}-\frac{59\!\cdots\!66}{50\!\cdots\!13}a^{6}+\frac{11\!\cdots\!48}{50\!\cdots\!13}a^{5}-\frac{15\!\cdots\!92}{50\!\cdots\!13}a^{4}+\frac{14\!\cdots\!00}{50\!\cdots\!13}a^{3}-\frac{20\!\cdots\!46}{50\!\cdots\!13}a^{2}-\frac{31\!\cdots\!75}{50\!\cdots\!13}a+\frac{17\!\cdots\!17}{39\!\cdots\!19}$ 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:  $37$
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^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127)
 
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^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127, 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^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127);
 
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^38 - x^37 - 222*x^36 + 479*x^35 + 21047*x^34 - 66053*x^33 - 1110228*x^32 + 4495506*x^31 + 35823405*x^30 - 180946047*x^29 - 731415663*x^28 + 4712734359*x^27 + 9238434306*x^26 - 83665939703*x^25 - 59973164681*x^24 + 1044644450973*x^23 - 101098208429*x^22 - 9336414182461*x^21 + 6288159462034*x^20 + 60150648390199*x^19 - 67252981484387*x^18 - 278611442416743*x^17 + 417500804951451*x^16 + 915079337067075*x^15 - 1716002791423036*x^14 - 2065408719688167*x^13 + 4825202788269126*x^12 + 2982003177766905*x^11 - 9269437748788275*x^10 - 2187058821094830*x^9 + 11855288436035595*x^8 - 411608485430453*x^7 - 9515695267587717*x^6 + 2357948299236310*x^5 + 4196479083397002*x^4 - 1892165812751211*x^3 - 650631784472963*x^2 + 526610168112801*x - 83497743723127);
 
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

$C_{38}$ (as 38T1):

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 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{457}) \), 19.19.755947441066272696677489606263668936388276269649.1

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 $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $38$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $38$ $38$ $38$ $19^{2}$ $38$ $38$

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
\(457\) Copy content Toggle raw display Deg $38$$38$$1$$37$