Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 3)
gp: K = bnfinit(y^34 - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^34 - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^34 - 3)
\( x^{34} - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(65355867734067162277675143712675696467755092714334768555947173347328\)
\(\medspace = 2^{34}\cdot 3^{33}\cdot 17^{34}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(98.76\) | 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: | $2\cdot 3^{33/34}17^{287/272}\approx 115.45768163941001$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$
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: | $17$ | 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: |
$a^{17}+2$, $a^{18}+a^{2}+1$, $a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $a^{32}+a^{30}+a^{28}-a^{24}-a^{22}-a^{20}-a^{16}-a^{14}-2a^{12}+a^{8}+3a^{6}+2a^{4}+2a^{2}+1$, $2a^{32}-a^{28}+a^{26}+3a^{24}-2a^{20}-a^{18}+3a^{16}-3a^{12}-3a^{10}+3a^{8}+2a^{6}-2a^{4}-3a^{2}+4$, $2a^{32}+3a^{30}+3a^{28}+3a^{26}+3a^{24}+a^{22}-a^{18}-3a^{16}-4a^{14}-5a^{12}-6a^{10}-4a^{8}-3a^{6}-2a^{4}+2a^{2}+4$, $a^{30}+a^{24}-a^{20}-a^{18}+2a^{16}+3a^{14}-a^{12}-3a^{10}-a^{8}+4a^{6}+3a^{4}-2a^{2}-2$, $a^{32}+a^{30}-3a^{28}+4a^{26}-4a^{24}+3a^{22}-a^{20}-2a^{18}+4a^{16}-5a^{14}+4a^{12}-2a^{10}-2a^{8}+6a^{6}-9a^{4}+9a^{2}-7$, $a^{30}-a^{28}+2a^{22}-a^{20}-a^{18}-a^{16}+2a^{14}-a^{10}+2a^{6}+a^{4}-3a^{2}-1$, $3a^{32}-3a^{30}+3a^{28}-3a^{26}+3a^{24}-2a^{22}+2a^{20}-2a^{18}+a^{16}+a^{14}-2a^{12}+3a^{10}-3a^{8}+4a^{6}-6a^{4}+8a^{2}-8$, $8a^{33}-7a^{32}+6a^{31}-6a^{30}+6a^{29}-8a^{28}+9a^{27}-10a^{26}+9a^{25}-8a^{24}+6a^{23}-5a^{22}+7a^{21}-11a^{20}+15a^{19}-16a^{18}+15a^{17}-12a^{16}+8a^{15}-5a^{14}+5a^{13}-9a^{12}+14a^{11}-19a^{10}+22a^{9}-21a^{8}+17a^{7}-12a^{6}+11a^{5}-10a^{4}+12a^{3}-15a^{2}+22a-25$, $a^{32}-a^{31}-3a^{30}+6a^{28}-5a^{26}-4a^{25}+7a^{24}+7a^{23}-6a^{22}-11a^{21}+2a^{20}+15a^{19}+a^{18}-12a^{17}-7a^{16}+11a^{15}+11a^{14}-8a^{13}-14a^{12}+a^{11}+18a^{10}+a^{9}-12a^{8}-6a^{7}+9a^{6}+6a^{5}-6a^{4}-5a^{3}+7a-2$, $4a^{32}-4a^{31}+4a^{30}+a^{29}-2a^{28}+4a^{27}-3a^{26}+4a^{25}+a^{24}-3a^{23}+9a^{22}-5a^{21}-a^{20}+9a^{19}-8a^{18}+5a^{17}+5a^{16}-7a^{15}+8a^{14}-5a^{13}+9a^{11}-12a^{10}+10a^{9}+3a^{8}-13a^{7}+13a^{6}-5a^{5}-2a^{4}+8a^{3}-6a^{2}+9a-7$, $20a^{32}-3a^{31}-13a^{30}-5a^{29}+30a^{28}-19a^{27}-11a^{26}+5a^{25}+23a^{24}-12a^{23}-26a^{22}+28a^{21}+4a^{20}-9a^{19}-28a^{18}+40a^{17}+6a^{16}-35a^{15}-2a^{14}+27a^{13}+15a^{12}-59a^{11}+31a^{10}+28a^{9}-12a^{8}-48a^{7}+28a^{6}+49a^{5}-58a^{4}-4a^{3}+27a^{2}+41a-77$, $22a^{33}+9a^{32}+20a^{31}+21a^{30}-12a^{29}-24a^{28}-13a^{27}-30a^{26}-38a^{25}+21a^{23}+9a^{22}+29a^{21}+49a^{20}+15a^{19}-9a^{18}+2a^{17}-23a^{16}-60a^{15}-30a^{14}+4a^{13}-7a^{12}+19a^{11}+70a^{10}+48a^{9}+11a^{8}+31a^{7}+7a^{6}-66a^{5}-64a^{4}-28a^{3}-51a^{2}-33a+56$, $65a^{33}+70a^{32}-66a^{30}-62a^{29}-44a^{28}-33a^{27}+19a^{26}+95a^{25}+96a^{24}+19a^{23}-43a^{22}-57a^{21}-83a^{20}-88a^{19}+2a^{18}+116a^{17}+113a^{16}+46a^{15}+14a^{14}-37a^{13}-140a^{12}-146a^{11}-13a^{10}+101a^{9}+106a^{8}+106a^{7}+101a^{6}-17a^{5}-188a^{4}-189a^{3}-48a^{2}+35a+77$, $36a^{33}+66a^{32}-8a^{31}-75a^{30}-27a^{29}+54a^{28}+71a^{27}-28a^{26}-85a^{25}-18a^{24}+72a^{23}+76a^{22}-55a^{21}-94a^{20}-3a^{19}+94a^{18}+73a^{17}-82a^{16}-105a^{15}+19a^{14}+120a^{13}+62a^{12}-114a^{11}-108a^{10}+40a^{9}+152a^{8}+41a^{7}-150a^{6}-104a^{5}+69a^{4}+177a^{3}+17a^{2}-192a-92$
| 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: | \( 520979448668698900000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{16}\cdot 520979448668698900000 \cdot 1}{2\cdot\sqrt{65355867734067162277675143712675696467755092714334768555947173347328}}\cr\approx \mathstrut & 0.760477183690835 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^34 - 3)
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^34 - 3, 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^34 - 3);
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^34 - 3);
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
$C_2\times F_{17}$ (as 34T9):
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 544 |
| The 34 conjugacy class representatives for $C_2\times F_{17}$ |
| Character table for $C_2\times F_{17}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 17.1.35609980753388072399570113617.1 |
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 34 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 | R | $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| Deg $16$ | $2$ | $8$ | $16$ | ||||
| Deg $16$ | $2$ | $8$ | $16$ | ||||
|
\(3\)
| Deg $34$ | $34$ | $1$ | $33$ | |||
|
\(17\)
| Deg $34$ | $17$ | $2$ | $34$ |