Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 3*x^17 + 3)
gp: K = bnfinit(y^34 - 3*y^17 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^34 - 3*x^17 + 3);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^34 - 3*x^17 + 3)
\( x^{34} - 3x^{17} + 3 \)
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: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3804212187770006845104225428814283599364285631706866468067\)
\(\medspace = -\,3^{33}\cdot 17^{34}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.38\) | 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: | $3^{33/34}17^{287/272}\approx 57.728840819705006$ | ||
| Ramified primes: |
\(3\), \(17\)
| 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: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( a^{17} - 1 \)
(order $6$)
| 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: |
$a^{17}+a-2$, $a-1$, $a^{33}+a^{30}+a^{27}+a^{24}+a^{21}+a^{18}-a^{17}-2a^{16}-a^{14}-2a^{13}-a^{11}-2a^{10}-a^{8}-2a^{7}-a^{5}-2a^{4}-a^{2}-2a+1$, $a^{33}+a^{32}-a^{30}-a^{29}+a^{27}+a^{26}-a^{24}-a^{23}+a^{21}+a^{20}-a^{18}-a^{16}-2a^{15}-a^{14}+a^{13}+2a^{12}+a^{11}-a^{10}-2a^{9}-a^{8}+a^{7}+2a^{6}+a^{5}-a^{4}-2a^{3}-a^{2}+2a+1$, $a^{33}-a^{30}-a^{28}+a^{25}+a^{23}-a^{21}-a^{20}-a^{19}+a^{18}-a^{16}+a^{14}+2a^{13}+a^{11}-a^{10}-2a^{8}+a^{7}-a^{6}+a^{5}+a^{4}+2a^{3}+a^{2}-2a-1$, $a^{33}-a^{28}-a^{27}-a^{25}-a^{24}-a^{21}+a^{20}+a^{19}+a^{17}-a^{16}+a^{14}+a^{13}+2a^{11}+2a^{10}+a^{8}+2a^{7}-a^{6}-a^{5}+a^{4}-2a^{3}-2a^{2}-a-1$, $a^{32}+a^{30}+a^{28}+a^{25}+a^{23}+a^{21}-a^{20}+a^{19}+a^{17}-2a^{15}-2a^{13}-2a^{11}-a^{9}-a^{8}-a^{7}-a^{6}-a^{5}-2a^{4}+a^{3}-2a^{2}-2$, $a^{32}-a^{28}+a^{26}+a^{25}-a^{24}+a^{19}-a^{17}-2a^{15}+a^{13}+a^{11}-a^{9}-2a^{8}+2a^{7}+a^{6}-a^{5}-a^{4}+a^{3}-a^{2}+2$, $a^{33}+a^{32}-a^{31}+a^{29}-a^{28}-a^{27}+a^{26}+a^{25}-a^{24}+2a^{22}-a^{21}-a^{20}+2a^{19}+a^{18}-a^{17}-2a^{16}-a^{15}+2a^{14}-a^{13}-a^{12}+3a^{11}+a^{10}-2a^{9}+2a^{7}-a^{6}-2a^{5}+3a^{4}+a^{3}-3a^{2}+2$, $a^{32}-a^{30}+a^{29}+a^{27}-a^{25}+2a^{23}-a^{22}+a^{21}+a^{18}-a^{15}+2a^{13}-2a^{12}-a^{11}+a^{8}-4a^{6}+2a^{5}-a^{4}-a^{3}-a-1$, $a^{33}+a^{32}+a^{31}-a^{29}-2a^{28}-a^{27}+a^{25}+a^{24}+a^{23}-a^{21}-a^{20}-a^{19}-a^{18}-a^{16}-a^{15}-2a^{14}-a^{13}+a^{12}+3a^{11}+2a^{10}+a^{9}-a^{7}-2a^{6}-2a^{5}+a^{3}+3a^{2}+3a+1$, $a^{33}-a^{32}-a^{31}+2a^{30}+2a^{29}-a^{28}-a^{27}+a^{26}+2a^{25}-2a^{23}-a^{22}+a^{21}+a^{20}-3a^{19}-3a^{18}+a^{17}+2a^{15}-a^{14}-4a^{13}-a^{12}+3a^{11}-3a^{9}-a^{8}+2a^{7}+3a^{6}-a^{4}+a^{3}+5a^{2}+2a-4$, $a^{31}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}-a^{17}-a^{16}-a^{15}-2a^{14}-a^{13}-2a^{11}-3a^{9}+a^{8}-3a^{7}-2a^{5}-a^{3}-2a^{2}-a+1$, $a^{33}+a^{32}-a^{31}-a^{29}+a^{28}-a^{27}+a^{26}+2a^{24}-a^{23}-a^{22}-2a^{21}+a^{17}-2a^{16}-a^{15}+a^{14}+a^{12}+2a^{10}-a^{9}-a^{8}-4a^{7}+2a^{6}+4a^{4}-a^{3}+3a^{2}-1$, $a^{33}+a^{32}+a^{31}-a^{30}+2a^{26}-a^{25}-a^{24}-a^{23}-a^{22}+2a^{21}+a^{20}-a^{18}-2a^{16}-a^{15}-a^{14}+2a^{13}-2a^{11}-3a^{9}+2a^{8}+3a^{7}+a^{5}-3a^{4}-a^{3}+2a^{2}+2a-1$, $a^{33}+a^{32}-a^{31}+a^{29}-2a^{28}-a^{27}+a^{26}-a^{25}-2a^{24}+a^{23}+2a^{22}-2a^{21}+3a^{19}-a^{17}-2a^{16}-2a^{15}+a^{14}-2a^{12}+3a^{11}+2a^{10}-2a^{9}+2a^{8}+4a^{7}-a^{6}-3a^{5}+4a^{4}+a^{3}-6a^{2}+2$
| 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: | \( 6031882093114950.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{17}\cdot 6031882093114950.0 \cdot 1}{6\cdot\sqrt{3804212187770006845104225428814283599364285631706866468067}}\cr\approx \mathstrut & 0.604264372666734 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^34 - 3*x^17 + 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*x^17 + 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*x^17 + 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*x^17 + 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 | ${\href{/padicField/2.8.0.1}{8} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $34$ | $34$ | $1$ | $33$ | |||
|
\(17\)
| Deg $34$ | $17$ | $2$ | $34$ |