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: |
\(132348898008484421823305810524036745773855649343936849\)
\(\medspace = 3\cdot 521\cdot 2833\cdot 4591297\cdot 17894485991227\cdot 363797878228018421962279049\)
| 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: | $3^{1/2}521^{1/2}2833^{1/2}4591297^{1/2}17894485991227^{1/2}363797878228018421962279049^{1/2}\approx 3.637978807091713e+26$ | ||
| Ramified primes: |
\(3\), \(521\), \(2833\), \(4591297\), \(17894485991227\), \(363797878228018421962279049\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{13234\!\cdots\!36849}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $8a^{24}+6a^{23}-5a^{22}-11a^{21}-6a^{20}+3a^{19}+7a^{18}+4a^{17}-2a^{16}-8a^{15}-12a^{14}-7a^{13}+9a^{12}+18a^{11}+2a^{10}-22a^{9}-22a^{8}+a^{7}+16a^{6}+10a^{5}-8a^{3}-23a^{2}-27a+5$, $21a^{25}-10a^{24}-16a^{23}+28a^{22}-6a^{21}-30a^{20}+35a^{19}+a^{18}-46a^{17}+44a^{16}+11a^{15}-62a^{14}+53a^{13}+21a^{12}-77a^{11}+55a^{10}+31a^{9}-88a^{8}+46a^{7}+46a^{6}-90a^{5}+25a^{4}+73a^{3}-82a^{2}-13a+7$, $14a^{25}+39a^{24}+62a^{23}+27a^{22}+21a^{21}+67a^{20}+63a^{19}+17a^{18}+52a^{17}+96a^{16}+51a^{15}+25a^{14}+101a^{13}+103a^{12}+29a^{11}+60a^{10}+146a^{9}+79a^{8}+26a^{7}+133a^{6}+170a^{5}+46a^{4}+78a^{3}+223a^{2}+147a-40$, $12a^{25}-12a^{24}-27a^{23}+6a^{22}+39a^{21}+31a^{20}-15a^{19}-39a^{18}-3a^{17}+32a^{16}+22a^{15}-48a^{14}-56a^{13}-12a^{12}+72a^{11}+37a^{10}-18a^{9}-69a^{8}+30a^{7}+75a^{6}+62a^{5}-91a^{4}-100a^{3}-29a^{2}+113a-20$, $12a^{25}+9a^{24}-6a^{23}-25a^{22}-21a^{21}+13a^{20}+40a^{19}+24a^{18}-24a^{17}-47a^{16}-21a^{15}+24a^{14}+48a^{13}+30a^{12}-19a^{11}-59a^{10}-46a^{9}+23a^{8}+79a^{7}+47a^{6}-37a^{5}-76a^{4}-35a^{3}+22a^{2}+49a-11$, $61a^{25}+90a^{24}+80a^{23}+63a^{22}+82a^{21}+58a^{20}+14a^{19}+21a^{18}-10a^{17}-75a^{16}-71a^{15}-93a^{14}-163a^{13}-146a^{12}-133a^{11}-190a^{10}-148a^{9}-83a^{8}-116a^{7}-49a^{6}+68a^{5}+47a^{4}+113a^{3}+262a^{2}+228a-58$, $7a^{24}+7a^{23}-a^{22}-6a^{21}-6a^{20}-4a^{19}+8a^{18}+15a^{17}-11a^{15}-8a^{14}-7a^{13}+4a^{12}+20a^{11}+9a^{10}-13a^{9}-17a^{8}-9a^{7}+9a^{6}+23a^{5}+14a^{4}-7a^{3}-26a^{2}-25a+5$, $248a^{25}-210a^{24}-305a^{23}+207a^{22}+349a^{21}-208a^{20}-412a^{19}+180a^{18}+457a^{17}-149a^{16}-530a^{15}+79a^{14}+582a^{13}-a^{12}-674a^{11}-124a^{10}+741a^{9}+254a^{8}-860a^{7}-436a^{6}+945a^{5}+612a^{4}-1085a^{3}-842a^{2}+1171a-194$, $5a^{25}-a^{24}+5a^{23}+14a^{22}+5a^{21}+19a^{20}+15a^{19}-11a^{18}-4a^{17}-14a^{16}-20a^{15}-19a^{13}-6a^{12}+28a^{11}+11a^{10}+34a^{9}+33a^{8}-15a^{7}+18a^{6}+8a^{5}-37a^{4}-12a^{3}-46a^{2}-34a+8$, $17a^{25}+15a^{24}-44a^{23}+70a^{22}-48a^{21}+24a^{20}+45a^{19}-60a^{18}+87a^{17}-53a^{16}+26a^{15}+27a^{14}-79a^{13}+114a^{12}-119a^{11}+47a^{10}+37a^{9}-162a^{8}+163a^{7}-160a^{6}+18a^{5}+61a^{4}-174a^{3}+207a^{2}-200a+34$, $65a^{25}+550a^{24}-584a^{23}-106a^{22}+749a^{21}-570a^{20}-315a^{19}+966a^{18}-528a^{17}-639a^{16}+1169a^{15}-356a^{14}-987a^{13}+1350a^{12}-101a^{11}-1478a^{10}+1442a^{9}+342a^{8}-1945a^{7}+1454a^{6}+910a^{5}-2526a^{4}+1229a^{3}+1724a^{2}-2981a+522$, $88a^{25}-36a^{24}-108a^{23}+42a^{22}+140a^{21}-40a^{20}-186a^{19}+18a^{18}+232a^{17}+22a^{16}-274a^{15}-87a^{14}+296a^{13}+161a^{12}-295a^{11}-234a^{10}+273a^{9}+296a^{8}-234a^{7}-330a^{6}+205a^{5}+355a^{4}-193a^{3}-369a^{2}+220a-29$
| 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: | \( 41285638269726584 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 41285638269726584 \cdot 1}{2\cdot\sqrt{132348898008484421823305810524036745773855649343936849}}\cr\approx \mathstrut & 0.859265261254915 \end{aligned}\] (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 $S_{26}$ |
| Character table for $S_{26}$ |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $17{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 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 | $[\ ]^{20}$ | |
|
\(521\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(2833\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(4591297\)
| $\Q_{4591297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{4591297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(17894485991227\)
| $\Q_{17894485991227}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17894485991227}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(363\!\cdots\!049\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |