Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x + 3)
gp: K = bnfinit(y^28 - 2*y + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 2*x + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 2*x + 3)
\( x^{28} - 2x + 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(252754298659611839720841922043746138323258475768971264\)
\(\medspace = 2^{28}\cdot 3^{27}\cdot 13\cdot 6553\cdot 20327\cdot 42022097279\cdot 1696876609651\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(80.77\) | 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: |
\(2\), \(3\), \(13\), \(6553\), \(20327\), \(42022097279\), \(1696876609651\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{37042\!\cdots\!98261}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
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: | $13$ | 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: |
$a^{19}+a^{11}-2a^{10}-a^{2}+a+1$, $a^{24}+a^{23}-a^{21}-2a^{20}-a^{19}+a^{18}+2a^{17}+2a^{16}-2a^{14}-2a^{13}-a^{12}+a^{11}+2a^{10}+a^{9}-a^{7}-a^{6}+1$, $a^{24}+a^{23}+a^{22}-a^{21}-a^{20}-a^{19}+a^{18}+a^{17}+a^{16}-a^{12}-a^{11}-a^{10}+a^{9}+a^{8}+a^{7}-a^{6}-a^{5}-a^{4}-1$, $a^{21}+a^{18}-a^{17}+a^{15}-a^{14}+a^{13}+2a^{12}-a^{11}+a^{10}+a^{9}-2a^{8}+a^{7}-a^{5}+2a^{4}+a-1$, $2a^{27}+4a^{26}+a^{25}-2a^{24}-2a^{23}+a^{22}+2a^{21}+a^{20}+a^{19}+a^{18}-3a^{16}-a^{15}+4a^{14}+7a^{13}+2a^{12}-4a^{11}-3a^{10}+2a^{9}+5a^{8}+a^{7}-4a^{3}-3a^{2}+5a+4$, $4a^{27}+8a^{26}+6a^{25}+5a^{24}+4a^{23}-a^{22}-6a^{21}-7a^{20}-7a^{19}-7a^{18}-2a^{17}+2a^{16}+5a^{15}+9a^{14}+10a^{13}+4a^{12}+a^{11}+a^{10}-8a^{9}-13a^{8}-7a^{7}-3a^{6}-6a^{5}+3a^{4}+13a^{3}+9a^{2}+6a+2$, $a^{27}+2a^{26}+2a^{25}+2a^{24}+a^{23}+a^{22}+a^{21}+2a^{20}+a^{19}+a^{16}-a^{14}+a^{12}+a^{11}+a^{9}+a^{8}+2a^{7}+3a^{3}-a-4$, $9a^{27}+8a^{26}+7a^{25}+10a^{24}+8a^{23}+2a^{22}+3a^{20}+6a^{19}+2a^{18}-a^{17}+a^{16}-4a^{15}-9a^{14}-4a^{13}+a^{12}-2a^{11}-11a^{10}-12a^{9}-6a^{8}-11a^{7}-12a^{6}-2a^{5}-5a^{4}-16a^{3}-19a^{2}-10a-20$, $30a^{27}+6a^{26}-27a^{25}-32a^{24}+8a^{23}+35a^{22}+22a^{21}-26a^{20}-36a^{19}-2a^{18}+45a^{17}+31a^{16}-19a^{15}-53a^{14}-13a^{13}+42a^{12}+49a^{11}-14a^{10}-61a^{9}-33a^{8}+42a^{7}+65a^{6}-69a^{4}-57a^{3}+38a^{2}+80a-32$, $11a^{27}+8a^{26}-11a^{25}-17a^{24}+5a^{23}+13a^{22}+9a^{21}-6a^{20}-21a^{19}-a^{18}+16a^{17}+13a^{16}-26a^{14}-12a^{13}+24a^{12}+18a^{11}+a^{10}-25a^{9}-22a^{8}+25a^{7}+24a^{6}+4a^{5}-17a^{4}-36a^{3}+13a^{2}+42a-13$, $10a^{27}+4a^{25}-2a^{24}-10a^{23}-a^{22}-12a^{21}-12a^{20}-6a^{19}-16a^{18}-a^{17}-a^{16}-10a^{15}+10a^{14}+2a^{13}+8a^{12}+20a^{11}-2a^{10}+15a^{9}+16a^{8}-4a^{7}+16a^{6}-7a^{5}-9a^{4}+10a^{3}-26a^{2}-6a-23$, $2a^{27}-a^{26}-6a^{25}+7a^{24}-3a^{23}+7a^{21}-9a^{20}+5a^{19}+a^{18}-9a^{17}+11a^{16}-4a^{15}+12a^{13}-13a^{12}+5a^{11}+2a^{10}-16a^{9}+15a^{8}-7a^{7}-4a^{6}+20a^{5}-21a^{4}+7a^{3}+5a^{2}-23a+20$, $a^{27}-7a^{26}-8a^{25}-a^{24}-4a^{23}-13a^{22}-11a^{21}+2a^{20}+3a^{19}-11a^{18}-10a^{17}+6a^{16}+14a^{15}+a^{14}+14a^{12}+19a^{11}+5a^{10}-7a^{9}+a^{8}+4a^{7}+2a^{6}-8a^{5}-9a^{4}-7a^{3}-2a^{2}-4a-23$
| 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: | \( 10505974365221896 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 10505974365221896 \cdot 1}{2\cdot\sqrt{252754298659611839720841922043746138323258475768971264}}\cr\approx \mathstrut & 1.56161949812928 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 2*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^28 - 2*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^28 - 2*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^28 - 2*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 304888344611713860501504000000 |
| The 3718 conjugacy class representatives for $S_{28}$ |
| Character table for $S_{28}$ |
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 | R | $22{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | $27{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | $27{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $28$ | $23{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
| 2.12.12.32 | $x^{12} - 4 x^{10} - 4 x^{9} + 26 x^{8} + 40 x^{7} - 4 x^{6} - 40 x^{5} + 28 x^{4} + 72 x^{3} + 24 x^{2} - 16 x + 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
| 2.12.12.31 | $x^{12} + 2 x^{10} + 2 x^{9} + 6 x^{8} + 12 x^{7} + 32 x^{6} + 48 x^{5} + 76 x^{4} + 48 x^{3} + 40 x^{2} + 8 x + 8$ | $4$ | $3$ | $12$ | 12T205 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $27$ | $27$ | $1$ | $27$ | ||||
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
|
\(6553\)
| $\Q_{6553}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{6553}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{6553}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(20327\)
| $\Q_{20327}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{20327}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(42022097279\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(1696876609651\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |