Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 5*x - 1)
gp: K = bnfinit(y^26 - 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 5*x - 1)
\( x^{26} - 5x - 1 \)
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: |
\(132348898008484434135544970938351367367204226144344401\)
\(\medspace = 17\cdot 3761\cdot 2736736843085821\cdot 75\!\cdots\!13\)
| 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: | $17^{1/2}3761^{1/2}2736736843085821^{1/2}756371290213139834941092507237413^{1/2}\approx 3.637978807091713e+26$ | ||
| Ramified primes: |
\(17\), \(3761\), \(2736736843085821\), \(75637\!\cdots\!37413\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{13234\!\cdots\!44401}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $6a^{25}+11a^{24}-7a^{23}+9a^{22}-27a^{21}+14a^{20}-12a^{19}+25a^{18}+3a^{17}-8a^{16}-a^{15}-35a^{14}+29a^{13}-14a^{12}+45a^{11}-19a^{10}-4a^{9}-19a^{8}-22a^{7}+47a^{6}-19a^{5}+57a^{4}-66a^{3}+9a^{2}-34a-7$, $10a^{25}-16a^{24}+9a^{23}-4a^{22}+9a^{21}-16a^{20}+7a^{19}+11a^{18}-17a^{17}+8a^{16}-5a^{15}+21a^{14}-30a^{13}+16a^{12}+10a^{11}-11a^{10}-a^{9}-4a^{8}+37a^{7}-51a^{6}+26a^{5}+2a^{4}+5a^{3}-26a^{2}+a$, $6a^{25}+16a^{24}+32a^{23}+31a^{22}+26a^{21}+37a^{20}+49a^{19}+34a^{18}+17a^{17}+27a^{16}+23a^{15}-9a^{14}-28a^{13}-28a^{12}-41a^{11}-75a^{10}-84a^{9}-80a^{8}-95a^{7}-92a^{6}-78a^{5}-69a^{4}-52a^{3}-5a^{2}+38a+7$, $25a^{25}-30a^{24}-58a^{23}-41a^{22}+26a^{21}+67a^{20}+57a^{19}-22a^{18}-80a^{17}-79a^{16}+14a^{15}+94a^{14}+109a^{13}+6a^{12}-101a^{11}-136a^{10}-24a^{9}+117a^{8}+177a^{7}+64a^{6}-123a^{5}-217a^{4}-114a^{3}+122a^{2}+256a+45$, $12a^{25}+a^{24}-3a^{23}-16a^{22}-4a^{21}+18a^{20}+14a^{19}+14a^{18}-4a^{17}-27a^{16}-4a^{15}+13a^{14}+23a^{13}+34a^{12}-8a^{11}-32a^{10}-11a^{9}-2a^{8}+43a^{7}+55a^{6}-7a^{5}-26a^{4}-37a^{3}-20a^{2}+69a+12$, $29a^{25}-a^{24}+15a^{23}+14a^{22}-26a^{21}-2a^{20}-22a^{19}-50a^{18}-3a^{17}-23a^{16}-7a^{15}+61a^{14}+25a^{13}+51a^{12}+71a^{11}-20a^{10}+a^{9}-29a^{8}-114a^{7}-33a^{6}-54a^{5}-62a^{4}+73a^{3}+18a^{2}+48a+8$, $231a^{25}-210a^{24}+183a^{23}-149a^{22}+105a^{21}-56a^{20}-5a^{19}+68a^{18}-146a^{17}+221a^{16}-309a^{15}+392a^{14}-484a^{13}+569a^{12}-654a^{11}+735a^{10}-807a^{9}+876a^{8}-926a^{7}+978a^{6}-1002a^{5}+1029a^{4}-1018a^{3}+1013a^{2}-966a-245$, $5a^{25}-3a^{24}-8a^{23}-5a^{22}+a^{21}+6a^{20}+12a^{19}+14a^{18}+3a^{17}-13a^{16}-20a^{15}-22a^{14}-20a^{13}-a^{12}+26a^{11}+37a^{10}+33a^{9}+21a^{8}-7a^{7}-41a^{6}-50a^{5}-28a^{4}+a^{3}+24a^{2}+40a+6$, $6a^{25}-3a^{24}-a^{23}+9a^{22}-9a^{21}+7a^{20}-7a^{19}+a^{18}-5a^{17}+5a^{16}-2a^{15}+5a^{14}+10a^{13}-15a^{12}+19a^{11}-22a^{10}+6a^{9}+2a^{8}-4a^{7}+13a^{6}-a^{5}+9a^{4}-16a^{3}+14a^{2}-39a-9$, $3a^{25}+4a^{24}+7a^{23}+4a^{22}-4a^{21}-6a^{20}-4a^{19}-3a^{18}-6a^{17}-15a^{16}-19a^{15}-12a^{14}-6a^{13}-8a^{12}-14a^{11}-15a^{10}-2a^{9}+13a^{8}+14a^{7}+8a^{6}+8a^{5}+24a^{4}+44a^{3}+39a^{2}+23a+2$, $21a^{25}-5a^{24}+9a^{23}-13a^{21}-3a^{20}-22a^{19}+2a^{18}-25a^{17}+18a^{16}-2a^{15}+15a^{14}+26a^{13}+12a^{12}+20a^{11}-9a^{10}+13a^{9}-49a^{8}-7a^{7}-37a^{6}-32a^{5}+4a^{4}-8a^{3}+54a^{2}-a-2$, $37a^{25}-73a^{24}+22a^{23}+64a^{22}-78a^{21}+5a^{20}+96a^{19}-75a^{18}-21a^{17}+129a^{16}-64a^{15}-64a^{14}+148a^{13}-51a^{12}-126a^{11}+152a^{10}-23a^{9}-187a^{8}+157a^{7}+38a^{6}-241a^{5}+147a^{4}+120a^{3}-295a^{2}+103a+34$
| 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: | \( 42996736288967990 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 42996736288967990 \cdot 1}{2\cdot\sqrt{132348898008484434135544970938351367367204226144344401}}\cr\approx \mathstrut & 0.894877816810689 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 5*x - 1)
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 - 1, 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 - 1);
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 - 1);
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.23.0.1 | $x^{23} + 15 x^{2} + 16 x + 14$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(3761\)
| $\Q_{3761}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(2736736843085821\)
| $\Q_{2736736843085821}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(756\!\cdots\!413\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |