Properties

Label 38.0.323...184.1
Degree $38$
Signature $[0, 19]$
Discriminant $-3.231\times 10^{103}$
Root discriminant \(529.58\)
Ramified primes $2,19$
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 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761)
 
gp: K = bnfinit(y^38 + 342*y^36 + 52193*y^34 + 4714147*y^32 + 281630122*y^30 + 11775412830*y^28 + 355501314519*y^26 + 7880575876524*y^24 + 129205470814704*y^22 + 1567217753510932*y^20 + 13989387194635573*y^18 + 90967185355366872*y^16 + 424520449507516382*y^14 + 1392518175895156030*y^12 + 3118364594061856178*y^10 + 4572362150288538208*y^8 + 4128788345874228337*y^6 + 2095436497414876275*y^4 + 523953812519262731*y^2 + 49228485006254761, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761)
 

\( x^{38} + 342 x^{36} + 52193 x^{34} + 4714147 x^{32} + 281630122 x^{30} + 11775412830 x^{28} + 355501314519 x^{26} + 7880575876524 x^{24} + \cdots + 49\!\cdots\!61 \) 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:  $[0, 19]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-323\!\cdots\!184\) \(\medspace = -\,2^{38}\cdot 19^{72}\) 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:  \(529.58\)
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:  $2\cdot 19^{36/19}\approx 529.5799757230775$
Ramified primes:   \(2\), \(19\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1444=2^{2}\cdot 19^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1444}(1,·)$, $\chi_{1444}(1027,·)$, $\chi_{1444}(647,·)$, $\chi_{1444}(267,·)$, $\chi_{1444}(1293,·)$, $\chi_{1444}(913,·)$, $\chi_{1444}(533,·)$, $\chi_{1444}(153,·)$, $\chi_{1444}(1179,·)$, $\chi_{1444}(799,·)$, $\chi_{1444}(419,·)$, $\chi_{1444}(39,·)$, $\chi_{1444}(1065,·)$, $\chi_{1444}(685,·)$, $\chi_{1444}(305,·)$, $\chi_{1444}(1331,·)$, $\chi_{1444}(951,·)$, $\chi_{1444}(571,·)$, $\chi_{1444}(191,·)$, $\chi_{1444}(1217,·)$, $\chi_{1444}(837,·)$, $\chi_{1444}(457,·)$, $\chi_{1444}(77,·)$, $\chi_{1444}(1103,·)$, $\chi_{1444}(723,·)$, $\chi_{1444}(343,·)$, $\chi_{1444}(1369,·)$, $\chi_{1444}(989,·)$, $\chi_{1444}(609,·)$, $\chi_{1444}(229,·)$, $\chi_{1444}(1255,·)$, $\chi_{1444}(875,·)$, $\chi_{1444}(495,·)$, $\chi_{1444}(115,·)$, $\chi_{1444}(1141,·)$, $\chi_{1444}(761,·)$, $\chi_{1444}(381,·)$, $\chi_{1444}(1407,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{23\!\cdots\!21}a^{36}-\frac{65\!\cdots\!98}{23\!\cdots\!21}a^{34}+\frac{56\!\cdots\!92}{23\!\cdots\!21}a^{32}-\frac{89\!\cdots\!99}{23\!\cdots\!21}a^{30}+\frac{54\!\cdots\!43}{23\!\cdots\!21}a^{28}+\frac{10\!\cdots\!90}{23\!\cdots\!21}a^{26}+\frac{67\!\cdots\!05}{23\!\cdots\!21}a^{24}-\frac{80\!\cdots\!41}{23\!\cdots\!21}a^{22}+\frac{31\!\cdots\!60}{23\!\cdots\!21}a^{20}-\frac{70\!\cdots\!10}{23\!\cdots\!21}a^{18}-\frac{72\!\cdots\!66}{23\!\cdots\!21}a^{16}+\frac{89\!\cdots\!40}{23\!\cdots\!21}a^{14}-\frac{50\!\cdots\!57}{23\!\cdots\!21}a^{12}+\frac{94\!\cdots\!95}{23\!\cdots\!21}a^{10}+\frac{72\!\cdots\!46}{23\!\cdots\!21}a^{8}-\frac{11\!\cdots\!97}{23\!\cdots\!21}a^{6}+\frac{10\!\cdots\!67}{23\!\cdots\!21}a^{4}+\frac{11\!\cdots\!86}{23\!\cdots\!21}a^{2}+\frac{27\!\cdots\!75}{23\!\cdots\!21}$, $\frac{1}{51\!\cdots\!51}a^{37}-\frac{18\!\cdots\!83}{51\!\cdots\!51}a^{35}+\frac{17\!\cdots\!82}{51\!\cdots\!51}a^{33}+\frac{23\!\cdots\!84}{51\!\cdots\!51}a^{31}+\frac{16\!\cdots\!08}{51\!\cdots\!51}a^{29}-\frac{18\!\cdots\!49}{51\!\cdots\!51}a^{27}+\frac{10\!\cdots\!87}{51\!\cdots\!51}a^{25}+\frac{11\!\cdots\!64}{51\!\cdots\!51}a^{23}-\frac{13\!\cdots\!59}{51\!\cdots\!51}a^{21}-\frac{17\!\cdots\!44}{51\!\cdots\!51}a^{19}-\frac{14\!\cdots\!27}{51\!\cdots\!51}a^{17}+\frac{87\!\cdots\!81}{51\!\cdots\!51}a^{15}+\frac{11\!\cdots\!57}{51\!\cdots\!51}a^{13}-\frac{17\!\cdots\!96}{51\!\cdots\!51}a^{11}+\frac{38\!\cdots\!79}{51\!\cdots\!51}a^{9}-\frac{22\!\cdots\!20}{51\!\cdots\!51}a^{7}+\frac{98\!\cdots\!34}{51\!\cdots\!51}a^{5}+\frac{18\!\cdots\!08}{51\!\cdots\!51}a^{3}+\frac{24\!\cdots\!69}{51\!\cdots\!51}a$ 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:  $18$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{2120994178005994038139395998259360021984270135379264159388192417778185970718250764}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{37} + \frac{723180419607108729511670586833785453607515728804356428164775673160068420495942350963}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{35} + \frac{109951150415306488393762655234911514552717898593438112456144425013770135941779134929171}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{33} + \frac{9884685807696188687556739989979566920355324601051079485828844940763734829543710043101429}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{31} + \frac{587090935121300643574067797425787038664071326912665511049500509744365425200035867526408117}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{29} + \frac{24367392769422748496951445425065602526824770484744124398838487178846401770315817267825859143}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{27} + \frac{728792811373915700062535197950702817192539038091338417307612957138877476036230182163484177183}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{25} + \frac{15961196015267782564479774289678301303516807692440887873888688053427343799885905256776896793047}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{23} + \frac{257575237183024110499155767668749200345212041405794336216089358835198310325616746779716659651750}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{21} + \frac{3059134881998184444287994571142540667611708656414932756748659570571260492038972745561632657639011}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{19} + \frac{26541119696363100115594341976293945866446534802820217996544274177244400508108690231124089883310574}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{17} + \frac{166010019712432072862507542924781309910646634047778432612428008034523729709726707781670985560911435}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{15} + \frac{734283566967917449676638898998672847161964847465108322005065000200076352259186993794528396544112895}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{13} + \frac{2235526409163826063268043933165353216814439619994479620140767148632472086684729142164078868751869841}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{11} + \frac{4510514232777519396481366335082287796742963350552013183743728953333472667508705879907088187663449789}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{9} + \frac{5715379493882463073511413729838406649414433916676289309553994043829376016691870531738477195069674173}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{7} + \frac{4205229931510312629937111548629401955649447567136371208220079457297172267371022520624834694606109281}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{5} + \frac{1575401593927194632160493072932916472587403547987819181396771418385217349765660025228879037282112922}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a^{3} + \frac{213835779666348379483519166482845437602423176537750505488365901711884738346524724267984462103710944}{5794081219884873070459632850776371640434768716111607348140342942902365685339904801762967343255559} a \)  (order $4$) 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 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761)
 
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 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761, 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 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761);
 
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 + 342*x^36 + 52193*x^34 + 4714147*x^32 + 281630122*x^30 + 11775412830*x^28 + 355501314519*x^26 + 7880575876524*x^24 + 129205470814704*x^22 + 1567217753510932*x^20 + 13989387194635573*x^18 + 90967185355366872*x^16 + 424520449507516382*x^14 + 1392518175895156030*x^12 + 3118364594061856178*x^10 + 4572362150288538208*x^8 + 4128788345874228337*x^6 + 2095436497414876275*x^4 + 523953812519262731*x^2 + 49228485006254761);
 
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

\(\Q(\sqrt{-1}) \), 19.19.10842505080063916320800450434338728415281531281.1

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

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
\(2\) Copy content Toggle raw display Deg $38$$2$$19$$38$
\(19\) Copy content Toggle raw display Deg $38$$19$$2$$72$