Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^38 + 2)
gp: K = bnfinit(y^38 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 + 2)
\( x^{38} + 2 \)
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: |
\(-147872192092304070375243338775640519188126273799287256562376917267775488\)
\(\medspace = -\,2^{75}\cdot 19^{38}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.63\) | 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: | $2^{75/38}19^{359/342}\approx 86.388418950132$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$a^{2}+1$, $a^{20}-a^{2}+1$, $a^{22}-a^{20}-a^{4}+a^{2}-1$, $a^{34}+a^{30}+a^{26}+a^{22}+a^{18}+a^{14}+a^{10}+a^{6}+a^{2}-1$, $a^{34}+a^{32}-a^{30}-a^{28}+a^{24}+a^{22}-2a^{18}-a^{16}+a^{14}+a^{12}+a^{10}-a^{8}-2a^{6}+2a^{2}+1$, $a^{36}-a^{35}+a^{33}-a^{32}+a^{30}-a^{29}+a^{27}-a^{26}+a^{24}-a^{23}+a^{21}-a^{20}+a^{18}-a^{17}+a^{15}-a^{14}+a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{32}+a^{24}-a^{8}-a^{2}-1$, $a^{32}+a^{30}+a^{28}-a^{22}+a^{16}+a^{14}-2a^{8}-a^{6}-a^{4}+1$, $a^{34}-a^{32}+a^{30}+a^{24}+a^{20}+a^{18}-a^{16}+2a^{14}+a^{10}+a^{8}-2a^{6}+3a^{4}-a^{2}+1$, $a^{36}-a^{32}-2a^{30}-2a^{28}-a^{26}+a^{24}+3a^{22}+3a^{20}+2a^{18}-2a^{14}-3a^{12}-2a^{10}+a^{6}+a^{4}-1$, $a^{36}+a^{35}-a^{34}-a^{33}+a^{31}+a^{30}-a^{29}-a^{28}-a^{27}+2a^{26}-2a^{23}+a^{21}+2a^{20}-a^{19}-2a^{18}-a^{17}+2a^{16}+2a^{15}+a^{14}-3a^{13}-a^{12}+3a^{10}-2a^{7}+a^{5}-a^{3}-a^{2}+a+1$, $3a^{37}-a^{36}-a^{35}-2a^{34}-a^{33}+3a^{32}+2a^{31}-3a^{29}-2a^{28}+5a^{26}-a^{24}-3a^{23}-a^{22}+2a^{21}+2a^{20}+a^{19}-2a^{18}-4a^{16}+3a^{15}+a^{14}+3a^{13}-a^{12}-4a^{11}-3a^{10}+4a^{9}+3a^{8}+2a^{7}-3a^{6}-6a^{5}+2a^{4}+2a^{3}+5a^{2}-2a-1$, $2a^{37}+a^{35}-2a^{34}-2a^{33}+2a^{31}+a^{30}+a^{29}+a^{28}-3a^{27}-a^{26}+2a^{24}+3a^{22}-2a^{21}-a^{20}-4a^{19}+3a^{18}-2a^{17}+4a^{16}+a^{14}-6a^{13}+2a^{12}-3a^{11}+4a^{10}+5a^{8}-5a^{7}-3a^{5}+2a^{4}-a^{3}+7a^{2}-a-1$, $5a^{37}-3a^{36}-6a^{34}+2a^{33}-2a^{32}+6a^{31}-2a^{30}+4a^{29}-6a^{28}+a^{27}-4a^{26}+7a^{25}-a^{24}+4a^{23}-7a^{22}-5a^{20}+6a^{19}+a^{18}+7a^{17}-3a^{16}-a^{15}-10a^{14}-a^{13}+a^{12}+10a^{11}+4a^{10}+2a^{9}-7a^{8}-7a^{7}-4a^{6}+a^{5}+7a^{4}+6a^{3}+6a^{2}-7a-5$, $3a^{37}+3a^{36}+5a^{34}+5a^{33}-5a^{32}-4a^{31}+2a^{30}-2a^{29}+7a^{27}+2a^{26}-5a^{25}+a^{23}-5a^{22}+a^{21}+5a^{20}-4a^{19}-4a^{18}+3a^{17}-a^{16}-3a^{15}+5a^{14}-11a^{12}-3a^{11}+3a^{10}-4a^{9}+2a^{8}+9a^{7}-4a^{6}-12a^{5}-2a^{4}-3a^{3}-8a^{2}+8a+11$, $16a^{37}-a^{36}-17a^{35}-a^{34}+14a^{33}-13a^{32}-13a^{31}+12a^{30}+11a^{29}-16a^{28}+22a^{26}+a^{25}-16a^{24}+10a^{23}+16a^{22}-18a^{21}-15a^{20}+15a^{19}+3a^{18}-25a^{17}+23a^{15}-8a^{14}-17a^{13}+19a^{12}+21a^{11}-19a^{10}-5a^{9}+26a^{8}+a^{7}-30a^{6}+6a^{5}+22a^{4}-21a^{3}-23a^{2}+20a+13$, $7a^{37}-3a^{36}+6a^{35}-3a^{34}-2a^{33}-3a^{32}-6a^{31}+a^{30}-2a^{29}+8a^{28}-a^{27}+7a^{26}-4a^{25}-a^{24}+2a^{23}-8a^{22}+7a^{21}-9a^{20}+4a^{19}-3a^{18}+7a^{16}-a^{15}+11a^{14}-5a^{13}+a^{12}-9a^{11}-3a^{10}-4a^{9}+8a^{7}-4a^{6}+14a^{5}-9a^{4}+6a^{3}-2a^{2}-4a+7$, $a^{37}-116a^{36}+139a^{35}-60a^{34}-75a^{33}+147a^{32}-112a^{31}-10a^{30}+133a^{29}-150a^{28}+61a^{27}+88a^{26}-172a^{25}+124a^{24}+20a^{23}-164a^{22}+177a^{21}-51a^{20}-119a^{19}+206a^{18}-122a^{17}-52a^{16}+193a^{15}-186a^{14}+29a^{13}+142a^{12}-221a^{11}+122a^{10}+68a^{9}-214a^{8}+208a^{7}-28a^{6}-175a^{5}+258a^{4}-136a^{3}-106a^{2}+265a-227$
| 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: | \( 584782896068373250000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{19}\cdot 584782896068373250000 \cdot 1}{2\cdot\sqrt{147872192092304070375243338775640519188126273799287256562376917267775488}}\cr\approx \mathstrut & 1.11285584041612 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^38 + 2)
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 + 2, 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 + 2);
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 + 2);
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
$C_2\times F_{19}$ (as 38T9):
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 solvable group of order 684 |
| The 38 conjugacy class representatives for $C_2\times F_{19}$ |
| Character table for $C_2\times F_{19}$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 19.1.518630842213417245507316350976.1 |
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]
Sibling fields
| Degree 38 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $18^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }^{6}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.9.0.1}{9} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | $18^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }^{6}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{19}$ | $18^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $38$ | $38$ | $1$ | $75$ | |||
|
\(19\)
| 19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |
| 19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |