Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 3)
gp: K = bnfinit(y^29 + 3*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 3*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 3*x - 3)
\( x^{29} + 3x - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(61015212974565260245306654909443807495102740216734089597\)
\(\medspace = 3^{28}\cdot 119533\cdot 419161\cdot 53\!\cdots\!29\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(83.88\) | 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: | $3^{28/29}119533^{1/2}419161^{1/2}53232187366908496998206741390729^{1/2}\approx 4.7172652522059916e+21$ | ||
| Ramified primes: |
\(3\), \(119533\), \(419161\), \(53232\!\cdots\!90729\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{26671\!\cdots\!21677}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
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: | $14$ | 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: |
$a-1$, $a^{10}-a+1$, $a^{28}+2a^{27}-2a^{25}-a^{24}+2a^{23}+2a^{22}-2a^{21}-3a^{20}+2a^{19}+4a^{18}-2a^{17}-5a^{16}+2a^{15}+6a^{14}-2a^{13}-7a^{12}+2a^{11}+8a^{10}-2a^{9}-9a^{8}+2a^{7}+10a^{6}-a^{5}-10a^{4}+9a^{2}+a-5$, $a^{28}+2a^{27}-2a^{25}-3a^{24}-2a^{23}+a^{21}+2a^{20}+2a^{19}+2a^{18}-2a^{16}-2a^{15}+4a^{13}+5a^{12}+3a^{11}-2a^{9}-2a^{8}-a^{7}+2a^{5}+4a^{4}+4a^{3}-a^{2}-6a-4$, $10a^{28}+10a^{27}+7a^{26}+2a^{25}-3a^{24}-6a^{23}-7a^{22}-6a^{21}-2a^{20}+4a^{19}+9a^{18}+11a^{17}+9a^{16}+3a^{15}-4a^{14}-9a^{13}-12a^{12}-12a^{11}-7a^{10}+a^{9}+8a^{8}+12a^{7}+11a^{6}+5a^{5}-2a^{4}-8a^{3}-14a^{2}-16a+20$, $24a^{28}+22a^{27}+18a^{26}+14a^{25}+12a^{24}+13a^{23}+15a^{22}+17a^{21}+16a^{20}+14a^{19}+10a^{18}+7a^{17}+5a^{16}+7a^{15}+11a^{14}+14a^{13}+13a^{12}+8a^{11}+3a^{10}+2a^{8}+6a^{7}+10a^{6}+11a^{5}+9a^{4}+5a^{3}-a^{2}-4a+70$, $10a^{28}+11a^{27}+10a^{26}+7a^{25}+5a^{21}+12a^{20}+10a^{19}+9a^{18}+a^{17}-5a^{16}-4a^{15}-3a^{14}+6a^{13}+11a^{12}+10a^{11}+5a^{10}-4a^{9}-8a^{8}-6a^{7}+2a^{6}+8a^{5}+16a^{4}+10a^{3}-6a+13$, $3a^{28}+3a^{27}-a^{26}+a^{25}+2a^{24}-3a^{23}-2a^{22}+4a^{21}+4a^{18}-7a^{16}+a^{15}+2a^{14}-4a^{13}+2a^{12}+6a^{11}-4a^{10}-4a^{9}+5a^{8}-5a^{7}-6a^{6}+9a^{5}+5a^{4}-7a^{3}+3a^{2}+3a-5$, $9a^{28}+7a^{27}+5a^{26}+7a^{25}+3a^{24}+a^{23}+8a^{22}+6a^{21}-4a^{20}+7a^{18}+a^{17}+3a^{15}-7a^{14}-7a^{13}+7a^{12}-2a^{11}-16a^{10}-2a^{9}+4a^{8}-7a^{7}+a^{6}+3a^{5}-16a^{4}-5a^{3}+16a^{2}-6a+8$, $34a^{28}+31a^{27}+33a^{26}+29a^{25}+26a^{24}+28a^{23}+25a^{22}+20a^{21}+22a^{20}+23a^{19}+16a^{18}+18a^{17}+22a^{16}+14a^{15}+12a^{14}+18a^{13}+14a^{12}+7a^{11}+15a^{10}+17a^{9}+4a^{8}+11a^{7}+15a^{6}+3a^{5}+6a^{4}+13a^{3}+8a^{2}+a+115$, $a^{28}+6a^{27}+5a^{26}+2a^{25}+a^{24}+6a^{23}+4a^{22}-3a^{21}+5a^{20}+6a^{19}+a^{18}+a^{17}+3a^{16}+6a^{15}-4a^{14}-5a^{13}+4a^{12}-a^{11}-9a^{10}-2a^{9}+8a^{8}-5a^{7}-7a^{6}+4a^{5}+a^{4}-7a^{3}-12a^{2}+7a+5$, $2a^{27}-2a^{26}-a^{25}+a^{24}-2a^{23}+4a^{22}+5a^{21}-a^{20}+5a^{19}-2a^{18}-5a^{17}+2a^{16}-4a^{15}-2a^{14}+5a^{13}-5a^{12}+a^{11}-5a^{9}+a^{8}+2a^{7}+2a^{6}+6a^{5}+3a^{4}+2a^{3}-6a^{2}-a-2$, $31a^{28}+33a^{27}+32a^{26}+32a^{25}+27a^{24}+24a^{23}+18a^{22}+15a^{21}+13a^{20}+13a^{19}+17a^{18}+17a^{17}+23a^{16}+19a^{15}+20a^{14}+14a^{13}+10a^{12}+5a^{11}+3a^{10}+4a^{9}+4a^{8}+14a^{7}+10a^{6}+20a^{5}+12a^{4}+13a^{3}+5a^{2}+91$, $a^{27}-7a^{26}+2a^{25}-9a^{24}+2a^{23}-3a^{22}-8a^{21}-10a^{19}+9a^{18}-11a^{17}-6a^{16}+9a^{15}-9a^{14}+9a^{13}-12a^{12}+6a^{11}+5a^{10}-10a^{9}+12a^{8}-22a^{7}+17a^{6}-2a^{5}-18a^{4}+12a^{3}-18a^{2}+26a-25$
| 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: | \( 34353828610448830 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 34353828610448830 \cdot 1}{2\cdot\sqrt{61015212974565260245306654909443807495102740216734089597}}\cr\approx \mathstrut & 0.657316048661033 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 3*x - 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^29 + 3*x - 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^29 + 3*x - 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^29 + 3*x - 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
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 non-solvable group of order 8841761993739701954543616000000 |
| The 4565 conjugacy class representatives for $S_{29}$ are not computed |
| Character table for $S_{29}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | $28{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $29$ | $29$ | $1$ | $28$ | |||
|
\(119533\)
| $\Q_{119533}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(419161\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(532\!\cdots\!729\)
| $\Q_{53\!\cdots\!29}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |