Properties

Label 38.0.223...259.1
Degree $38$
Signature $[0, 19]$
Discriminant $-2.234\times 10^{93}$
Root discriminant \(286.12\)
Ramified prime $19$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023)
 
gp: K = bnfinit(y^38 - 209*y^35 + 190*y^34 + 1292*y^33 + 11818*y^32 - 17860*y^31 - 124165*y^30 + 92378*y^29 - 315590*y^28 + 461434*y^27 + 19355661*y^26 + 42336256*y^25 - 170675214*y^24 - 2184021538*y^23 + 4055379323*y^22 + 20028771815*y^21 - 37281886089*y^20 - 119372774190*y^19 + 206222273796*y^18 + 638565435242*y^17 - 1510749792677*y^16 - 1246896758667*y^15 + 6615154002166*y^14 + 1497131351993*y^13 + 895955914113*y^12 - 6914578071407*y^11 + 51068057863185*y^10 + 80278712330599*y^9 - 10972845112675*y^8 + 78343381011053*y^7 + 257531197354487*y^6 + 248070995565179*y^5 + 172436840306532*y^4 + 97988777901672*y^3 + 65070800259551*y^2 + 24610290628033*y + 5454582062023, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023)
 

\( x^{38} - 209 x^{35} + 190 x^{34} + 1292 x^{33} + 11818 x^{32} - 17860 x^{31} - 124165 x^{30} + 92378 x^{29} - 315590 x^{28} + 461434 x^{27} + 19355661 x^{26} + \cdots + 5454582062023 \) 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:   \(-223\!\cdots\!259\) \(\medspace = -\,19^{73}\) 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:  \(286.12\)
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:  $19^{73/38}\approx 286.1231319470719$
Ramified primes:   \(19\) 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:  \(\Q(\sqrt{-19}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(361=19^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{361}(1,·)$, $\chi_{361}(132,·)$, $\chi_{361}(134,·)$, $\chi_{361}(265,·)$, $\chi_{361}(267,·)$, $\chi_{361}(18,·)$, $\chi_{361}(20,·)$, $\chi_{361}(151,·)$, $\chi_{361}(153,·)$, $\chi_{361}(284,·)$, $\chi_{361}(286,·)$, $\chi_{361}(37,·)$, $\chi_{361}(39,·)$, $\chi_{361}(170,·)$, $\chi_{361}(172,·)$, $\chi_{361}(303,·)$, $\chi_{361}(305,·)$, $\chi_{361}(56,·)$, $\chi_{361}(58,·)$, $\chi_{361}(189,·)$, $\chi_{361}(191,·)$, $\chi_{361}(322,·)$, $\chi_{361}(324,·)$, $\chi_{361}(75,·)$, $\chi_{361}(77,·)$, $\chi_{361}(208,·)$, $\chi_{361}(210,·)$, $\chi_{361}(341,·)$, $\chi_{361}(343,·)$, $\chi_{361}(94,·)$, $\chi_{361}(96,·)$, $\chi_{361}(227,·)$, $\chi_{361}(229,·)$, $\chi_{361}(360,·)$, $\chi_{361}(113,·)$, $\chi_{361}(115,·)$, $\chi_{361}(246,·)$, $\chi_{361}(248,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{1123845388013}a^{36}+\frac{81759373168}{1123845388013}a^{35}+\frac{218586334288}{1123845388013}a^{34}-\frac{457169512039}{1123845388013}a^{33}-\frac{182026071013}{1123845388013}a^{32}-\frac{537349195266}{1123845388013}a^{31}-\frac{301263568141}{1123845388013}a^{30}-\frac{348191628129}{1123845388013}a^{29}-\frac{431806770831}{1123845388013}a^{28}-\frac{57842106166}{1123845388013}a^{27}+\frac{236862564189}{1123845388013}a^{26}-\frac{77373154571}{1123845388013}a^{25}-\frac{229247270794}{1123845388013}a^{24}+\frac{158637833776}{1123845388013}a^{23}-\frac{104028860513}{1123845388013}a^{22}-\frac{420125099130}{1123845388013}a^{21}-\frac{439884885972}{1123845388013}a^{20}-\frac{445027056863}{1123845388013}a^{19}-\frac{271202263346}{1123845388013}a^{18}-\frac{394809801340}{1123845388013}a^{17}-\frac{93952947123}{1123845388013}a^{16}-\frac{94967709896}{1123845388013}a^{15}+\frac{437454545342}{1123845388013}a^{14}-\frac{58950789284}{1123845388013}a^{13}+\frac{308122839381}{1123845388013}a^{12}+\frac{464732530922}{1123845388013}a^{11}-\frac{98374494772}{1123845388013}a^{10}-\frac{363358723340}{1123845388013}a^{9}+\frac{50081863872}{1123845388013}a^{8}+\frac{408500857090}{1123845388013}a^{7}-\frac{165018752863}{1123845388013}a^{6}+\frac{140800715710}{1123845388013}a^{5}-\frac{251153399305}{1123845388013}a^{4}-\frac{485051111513}{1123845388013}a^{3}-\frac{393710228217}{1123845388013}a^{2}-\frac{469715890334}{1123845388013}a+\frac{499511957736}{1123845388013}$, $\frac{1}{19\!\cdots\!29}a^{37}+\frac{69\!\cdots\!47}{19\!\cdots\!29}a^{36}-\frac{20\!\cdots\!78}{19\!\cdots\!29}a^{35}-\frac{22\!\cdots\!42}{19\!\cdots\!29}a^{34}+\frac{56\!\cdots\!95}{19\!\cdots\!29}a^{33}-\frac{33\!\cdots\!41}{19\!\cdots\!29}a^{32}+\frac{84\!\cdots\!56}{19\!\cdots\!29}a^{31}+\frac{87\!\cdots\!41}{19\!\cdots\!29}a^{30}-\frac{42\!\cdots\!05}{19\!\cdots\!29}a^{29}-\frac{42\!\cdots\!46}{19\!\cdots\!29}a^{28}-\frac{50\!\cdots\!21}{19\!\cdots\!29}a^{27}+\frac{25\!\cdots\!78}{19\!\cdots\!29}a^{26}-\frac{82\!\cdots\!80}{19\!\cdots\!29}a^{25}-\frac{95\!\cdots\!96}{19\!\cdots\!29}a^{24}+\frac{90\!\cdots\!94}{19\!\cdots\!29}a^{23}-\frac{30\!\cdots\!48}{19\!\cdots\!29}a^{22}-\frac{12\!\cdots\!97}{19\!\cdots\!29}a^{21}+\frac{25\!\cdots\!82}{19\!\cdots\!29}a^{20}+\frac{25\!\cdots\!98}{19\!\cdots\!29}a^{19}-\frac{87\!\cdots\!62}{19\!\cdots\!29}a^{18}+\frac{56\!\cdots\!21}{19\!\cdots\!29}a^{17}+\frac{36\!\cdots\!64}{19\!\cdots\!29}a^{16}-\frac{63\!\cdots\!25}{19\!\cdots\!29}a^{15}-\frac{42\!\cdots\!85}{19\!\cdots\!29}a^{14}+\frac{81\!\cdots\!67}{19\!\cdots\!29}a^{13}+\frac{30\!\cdots\!64}{19\!\cdots\!29}a^{12}-\frac{49\!\cdots\!81}{19\!\cdots\!29}a^{11}+\frac{54\!\cdots\!74}{19\!\cdots\!29}a^{10}+\frac{14\!\cdots\!66}{19\!\cdots\!29}a^{9}-\frac{30\!\cdots\!09}{19\!\cdots\!29}a^{8}-\frac{14\!\cdots\!45}{19\!\cdots\!29}a^{7}+\frac{75\!\cdots\!19}{19\!\cdots\!29}a^{6}-\frac{88\!\cdots\!77}{19\!\cdots\!29}a^{5}+\frac{35\!\cdots\!41}{19\!\cdots\!29}a^{4}+\frac{81\!\cdots\!04}{19\!\cdots\!29}a^{3}+\frac{91\!\cdots\!60}{19\!\cdots\!29}a^{2}-\frac{18\!\cdots\!81}{19\!\cdots\!29}a+\frac{45\!\cdots\!02}{19\!\cdots\!29}$ 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:   \( -1 \)  (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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023)
 
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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023, 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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023);
 
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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023);
 
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

\(\Q(\sqrt{-19}) \), 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 $38$ $38$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ R $19^{2}$ $38$ $38$ $38$ $38$ $19^{2}$ $19^{2}$ $38$ $38$

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
\(19\) Copy content Toggle raw display Deg $38$$38$$1$$73$