Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 2*x - 4)
gp: K = bnfinit(y^29 + 2*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 2*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 2*x - 4)
\( x^{29} + 2x - 4 \)
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: |
\(46255321614773567947562404409632020776155679191255748706304\)
\(\medspace = 2^{54}\cdot 853\cdot 133051\cdot 22\!\cdots\!27\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.42\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(853\), \(133051\), \(22624\!\cdots\!32827\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{25676\!\cdots\!95981}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{2}a^{28}$
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
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: |
$a-1$, $a^{26}-a^{25}+2a^{23}-2a^{22}-a^{21}+2a^{20}-3a^{19}-a^{18}+4a^{17}-2a^{16}-a^{15}+4a^{14}-3a^{13}-2a^{12}+4a^{11}-3a^{10}-2a^{9}+4a^{8}-3a^{7}-a^{6}+5a^{5}-2a^{4}-a^{3}+a^{2}-4a+1$, $a^{28}-a^{27}+4a^{26}+a^{24}-a^{23}-5a^{22}+a^{21}-3a^{20}+4a^{19}+a^{18}-a^{17}+3a^{16}-6a^{15}+3a^{14}-3a^{13}+2a^{12}+6a^{11}-2a^{10}+6a^{9}-9a^{8}-2a^{7}-a^{6}-5a^{5}+12a^{4}-7a^{3}+8a^{2}-5a-3$, $6a^{28}-10a^{27}-22a^{26}-28a^{25}-30a^{24}-29a^{23}-27a^{22}-20a^{21}-3a^{20}+18a^{19}+36a^{18}+44a^{17}+39a^{16}+30a^{15}+22a^{14}+12a^{13}-4a^{12}-26a^{11}-49a^{10}-60a^{9}-52a^{8}-35a^{7}-13a^{6}+7a^{5}+20a^{4}+39a^{3}+57a^{2}+69a+79$, $91a^{28}+113a^{27}+122a^{26}+114a^{25}+74a^{24}+10a^{23}-59a^{22}-115a^{21}-140a^{20}-143a^{19}-130a^{18}-107a^{17}-58a^{16}+11a^{15}+99a^{14}+173a^{13}+201a^{12}+186a^{11}+135a^{10}+77a^{9}+14a^{8}-53a^{7}-144a^{6}-222a^{5}-263a^{4}-240a^{3}-144a^{2}-35a+249$, $a^{28}-4a^{26}-7a^{25}+a^{23}+6a^{22}+7a^{21}-2a^{20}-2a^{19}-8a^{18}-7a^{17}+5a^{16}+3a^{15}+8a^{14}+8a^{13}-9a^{12}-4a^{11}-10a^{10}-6a^{9}+13a^{8}+4a^{7}+13a^{6}-16a^{4}-4a^{3}-12a^{2}+2a+23$, $13a^{28}+7a^{27}+2a^{26}-25a^{25}+15a^{23}+13a^{22}-2a^{21}-32a^{20}+10a^{19}+14a^{18}+11a^{17}-9a^{16}-31a^{15}+19a^{14}+9a^{13}+20a^{12}-18a^{11}-39a^{10}+27a^{9}+17a^{8}+29a^{7}-49a^{6}-40a^{5}+46a^{4}+21a^{3}+34a^{2}-70a-3$, $7a^{28}+18a^{27}+24a^{26}+26a^{25}+29a^{24}+33a^{23}+34a^{22}+24a^{21}+4a^{20}-24a^{19}-50a^{18}-62a^{17}-56a^{16}-36a^{15}-10a^{14}+15a^{13}+28a^{12}+33a^{11}+37a^{10}+43a^{9}+51a^{8}+53a^{7}+40a^{6}+a^{5}-43a^{4}-83a^{3}-99a^{2}-88a-35$, $33a^{28}+10a^{27}-17a^{26}-29a^{25}+35a^{24}+26a^{23}-19a^{22}-40a^{21}+26a^{20}+39a^{19}-22a^{18}-48a^{17}+7a^{16}+60a^{15}-14a^{14}-50a^{13}-20a^{12}+90a^{11}-a^{10}-51a^{9}-54a^{8}+116a^{7}+9a^{6}-49a^{5}-97a^{4}+121a^{3}+26a^{2}-34a-77$, $4a^{26}+6a^{25}+3a^{24}+4a^{23}+4a^{22}-4a^{20}-4a^{19}-5a^{18}-10a^{17}-7a^{16}-3a^{15}-4a^{14}-a^{13}+9a^{12}+6a^{11}+7a^{10}+17a^{9}+8a^{8}+3a^{7}+10a^{6}+a^{5}-13a^{4}-a^{3}-7a^{2}-19a-3$, $7a^{28}+9a^{27}+2a^{26}+6a^{25}+10a^{24}+7a^{23}+12a^{22}+7a^{21}+7a^{20}+12a^{19}+3a^{18}+11a^{17}+13a^{16}+5a^{15}+10a^{14}+a^{13}+8a^{12}+20a^{11}+5a^{10}+12a^{9}+10a^{8}+6a^{7}+22a^{6}+7a^{5}+14a^{4}+23a^{3}-2a^{2}+12a+25$, $20a^{28}+21a^{27}-9a^{26}-23a^{25}-3a^{24}+28a^{23}+20a^{22}-15a^{21}-27a^{20}+a^{19}+40a^{18}+20a^{17}-24a^{16}-38a^{15}+9a^{14}+47a^{13}+27a^{12}-32a^{11}-47a^{10}+20a^{9}+54a^{8}+23a^{7}-50a^{6}-45a^{5}+27a^{4}+81a^{3}+18a^{2}-76a-11$, $2a^{27}+11a^{26}-5a^{24}+3a^{23}-4a^{22}-13a^{21}+4a^{20}+4a^{19}-4a^{18}+8a^{17}+13a^{16}-4a^{15}-7a^{14}+8a^{13}-14a^{12}-16a^{11}+8a^{10}+4a^{9}-13a^{8}+23a^{7}+13a^{6}-14a^{5}+3a^{4}+9a^{3}-27a^{2}-14a+19$, $64a^{28}+45a^{27}-4a^{26}-59a^{25}-73a^{24}-42a^{23}+29a^{22}+75a^{21}+82a^{20}+19a^{19}-48a^{18}-102a^{17}-73a^{16}-9a^{15}+87a^{14}+109a^{13}+76a^{12}-37a^{11}-110a^{10}-131a^{9}-41a^{8}+59a^{7}+148a^{6}+125a^{5}+25a^{4}-119a^{3}-178a^{2}-119a+157$
| 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: | \( 2017302844328075800 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 2017302844328075800 \cdot 1}{2\cdot\sqrt{46255321614773567947562404409632020776155679191255748706304}}\cr\approx \mathstrut & 1.40187214155992 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 2*x - 4)
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 + 2*x - 4, 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 + 2*x - 4);
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 + 2*x - 4);
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 | R | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $26{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $28$ | $28$ | $1$ | $54$ | ||||
|
\(853\)
| $\Q_{853}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(133051\)
| $\Q_{133051}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(226\!\cdots\!827\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |