Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 5*x - 2)
gp: K = bnfinit(y^26 - 5*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 5*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 5*x - 2)
\( x^{26} - 5x - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(132348898215049523817354996629623929806802964132075857\)
\(\medspace = 3\cdot 19\cdot 41\cdot 443\cdot 1481\cdot 8314099\cdot 221768083\cdot 2040445848379\cdot 22943703711693769\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(110.44\) | 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: | $3^{1/2}19^{1/2}41^{1/2}443^{1/2}1481^{1/2}8314099^{1/2}221768083^{1/2}2040445848379^{1/2}22943703711693769^{1/2}\approx 3.637978809930722e+26$ | ||
| Ramified primes: |
\(3\), \(19\), \(41\), \(443\), \(1481\), \(8314099\), \(221768083\), \(2040445848379\), \(22943703711693769\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{13234\!\cdots\!75857}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
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: |
$6a^{24}+9a^{23}+8a^{22}-3a^{21}-11a^{20}-14a^{19}-8a^{18}+7a^{17}+14a^{16}+15a^{15}+5a^{14}-9a^{13}-12a^{12}-13a^{11}-3a^{10}+4a^{9}+6a^{8}+12a^{7}+10a^{6}+12a^{5}+5a^{4}-15a^{3}-27a^{2}-36a-11$, $77a^{25}-23a^{24}+6a^{23}+4a^{22}-8a^{21}+6a^{20}-10a^{19}+8a^{18}-5a^{17}+10a^{16}-a^{15}+5a^{14}-3a^{13}-4a^{12}-3a^{11}-7a^{10}+7a^{9}-5a^{8}+17a^{7}-11a^{6}+14a^{5}-26a^{4}+10a^{3}-31a^{2}+23a-405$, $78a^{25}-30a^{24}+11a^{23}-4a^{22}+a^{20}+a^{18}-a^{17}-2a^{16}+3a^{15}-3a^{14}+4a^{13}-2a^{12}-a^{11}+a^{9}-2a^{7}+4a^{6}-6a^{5}+6a^{4}-3a^{3}-2a^{2}+a-391$, $3a^{25}-3a^{24}-3a^{23}+4a^{22}+2a^{21}-4a^{20}-a^{19}+5a^{18}-a^{17}-4a^{16}+6a^{15}+4a^{14}-9a^{13}-7a^{12}+6a^{11}+3a^{10}-5a^{9}+7a^{8}+9a^{7}-13a^{6}-15a^{5}+8a^{4}+9a^{3}-7a^{2}+4a+3$, $9a^{25}-6a^{24}-7a^{23}+13a^{22}-2a^{21}-11a^{20}+13a^{19}+a^{18}-16a^{17}+14a^{16}+8a^{15}-20a^{14}+8a^{13}+11a^{12}-26a^{11}+6a^{10}+25a^{9}-22a^{8}-4a^{7}+29a^{6}-24a^{5}-12a^{4}+43a^{3}-15a^{2}-33a-7$, $a^{25}-4a^{24}+a^{22}-3a^{21}-3a^{20}-a^{19}-7a^{16}-a^{15}+a^{14}-2a^{13}-5a^{12}-7a^{11}+3a^{10}-a^{9}-10a^{8}-6a^{7}-2a^{6}+a^{5}-10a^{4}-15a^{3}+2a^{2}-4a-13$, $8a^{25}+7a^{24}+2a^{23}-7a^{22}-5a^{21}-2a^{20}-a^{19}-10a^{18}-20a^{17}-18a^{16}-10a^{15}-4a^{14}-14a^{13}-20a^{12}-16a^{11}+5a^{10}+18a^{9}+10a^{8}+3a^{7}+9a^{6}+39a^{5}+52a^{4}+41a^{3}+17a^{2}+20a+5$, $4a^{25}+a^{24}-2a^{23}-6a^{21}-a^{20}+4a^{19}-5a^{18}-a^{17}+a^{16}-a^{15}+11a^{14}+3a^{13}-7a^{12}+3a^{11}-5a^{10}-3a^{9}+5a^{8}-18a^{7}-10a^{6}+11a^{5}+10a^{3}+6a^{2}-12a-5$, $17a^{25}-21a^{23}-5a^{22}+22a^{21}+9a^{20}-24a^{19}-14a^{18}+25a^{17}+20a^{16}-24a^{15}-25a^{14}+27a^{13}+36a^{12}-22a^{11}-43a^{10}+20a^{9}+56a^{8}-11a^{7}-63a^{6}+2a^{5}+70a^{4}+5a^{3}-84a^{2}-23a+3$, $23a^{25}+8a^{24}-14a^{23}-9a^{22}+18a^{21}+31a^{20}+12a^{19}-17a^{18}-14a^{17}+24a^{16}+45a^{15}+17a^{14}-21a^{13}-17a^{12}+28a^{11}+61a^{10}+34a^{9}-26a^{8}-32a^{7}+37a^{6}+91a^{5}+53a^{4}-32a^{3}-50a^{2}+44a+23$, $2a^{25}-a^{24}+12a^{23}+15a^{22}-3a^{21}+23a^{20}+19a^{19}+8a^{18}+29a^{17}+3a^{16}+a^{15}+27a^{14}-9a^{13}-8a^{12}-3a^{11}-45a^{10}-14a^{9}-16a^{8}-66a^{7}-30a^{6}-46a^{5}-58a^{4}+9a^{3}-39a^{2}-40a+27$, $40a^{25}-29a^{24}+21a^{23}-9a^{21}+7a^{20}+6a^{19}-19a^{18}-4a^{17}+23a^{16}+6a^{15}-17a^{14}-7a^{13}+7a^{12}+2a^{11}-4a^{10}+8a^{9}+12a^{8}-18a^{7}-22a^{6}+27a^{5}+18a^{4}-40a^{3}+5a^{2}+57a-233$, $1229a^{25}-491a^{24}+206a^{23}-78a^{22}+22a^{21}-17a^{20}+14a^{19}+6a^{18}-4a^{17}-6a^{16}+6a^{15}+5a^{14}-9a^{13}-7a^{12}+14a^{11}+17a^{10}-10a^{9}-24a^{8}+2a^{7}+26a^{6}+8a^{5}-17a^{4}-3a^{3}+19a^{2}-6171$
| 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: | \( 85063692836779780 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 85063692836779780 \cdot 1}{2\cdot\sqrt{132348898215049523817354996629623929806802964132075857}}\cr\approx \mathstrut & 1.77040441313123 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 5*x - 2)
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^26 - 5*x - 2, 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^26 - 5*x - 2);
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^26 - 5*x - 2);
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 403291461126605635584000000 |
| The 2436 conjugacy class representatives for $S_{26}$ are not computed |
| Character table for $S_{26}$ 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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $22{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | $24{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $15{,}\,{\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.20.0.1 | $x^{20} + 2 x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + 2 x^{5} + 2 x^{4} + 2 x^{3} + x + 2$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.5.0.1 | $x^{5} + 40 x^{2} + 14 x + 35$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 41.18.0.1 | $x^{18} + x^{11} + 7 x^{10} + 20 x^{9} + 23 x^{8} + 35 x^{7} + 38 x^{6} + 24 x^{5} + 12 x^{4} + 29 x^{3} + 10 x^{2} + 6 x + 6$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(443\)
| $\Q_{443}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(1481\)
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(8314099\)
| $\Q_{8314099}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(221768083\)
| $\Q_{221768083}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{221768083}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(2040445848379\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(22943703711693769\)
| $\Q_{22943703711693769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{22943703711693769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{22943703711693769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |