Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 4*x - 2)
gp: K = bnfinit(y^26 - 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 4*x - 2)
\( x^{26} - 4x - 2 \)
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: |
\(400000206565095837929605898429873236273026387935232\)
\(\medspace = 2^{51}\cdot 3\cdot 11\cdot 73\cdot 1409\cdot 10429777\cdot 5017733838435325202707\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(88.36\) | 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: |
\(2\), \(3\), \(11\), \(73\), \(1409\), \(10429777\), \(5017733838435325202707\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{35527\!\cdots\!94518}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{11}+a^{10}+a^{7}+a^{6}-a^{4}-a-1$, $6a^{25}-3a^{24}+a^{23}+a^{11}-a^{10}+a^{7}-a^{6}+2a^{3}-2a^{2}-23$, $a^{25}-a^{21}-a^{20}-a^{18}-a^{17}+a^{16}+a^{15}+a^{14}+2a^{13}+a^{12}+a^{10}-a^{9}-3a^{8}-2a^{7}-a^{6}-2a^{5}+2a^{3}+2a^{2}+3a+1$, $3a^{25}-3a^{24}+3a^{23}-2a^{22}+2a^{20}-3a^{19}+3a^{18}-2a^{17}+2a^{15}-3a^{14}+3a^{13}-2a^{12}+2a^{10}-3a^{9}+3a^{8}-2a^{7}+3a^{5}-6a^{4}+6a^{3}-3a^{2}-9$, $a^{25}+a^{24}-a^{22}-a^{21}+a^{19}+a^{18}-a^{16}-a^{15}+a^{13}+a^{12}-a^{10}-a^{9}+a^{7}+a^{6}-a^{4}-a^{3}+2a+1$, $2a^{25}+a^{24}+2a^{22}-2a^{21}-4a^{20}-2a^{19}+6a^{18}+3a^{17}-a^{16}-a^{15}-2a^{14}-3a^{13}+10a^{11}+4a^{10}-5a^{9}-6a^{8}-a^{7}+2a^{5}+12a^{4}+a^{3}-14a^{2}-13a-3$, $11a^{25}-6a^{24}+4a^{23}-a^{22}-a^{20}-3a^{19}-a^{18}+2a^{17}+a^{16}-3a^{14}-2a^{13}-2a^{12}-a^{11}+4a^{10}+a^{9}-3a^{8}-5a^{7}-4a^{6}+a^{5}+a^{4}+3a^{3}-9a-51$, $3a^{25}-6a^{24}+2a^{23}+3a^{22}-5a^{21}+7a^{20}+a^{19}-3a^{18}+7a^{17}-2a^{16}-6a^{15}+7a^{14}-11a^{13}-a^{12}+4a^{11}-8a^{10}+4a^{9}+10a^{8}-9a^{7}+15a^{6}+2a^{5}-7a^{4}+11a^{3}-9a^{2}-14a-3$, $2a^{25}+14a^{24}-26a^{23}+21a^{22}-a^{21}-21a^{20}+29a^{19}-14a^{18}-14a^{17}+33a^{16}-27a^{15}+30a^{13}-40a^{12}+18a^{11}+20a^{10}-45a^{9}+38a^{8}-a^{7}-42a^{6}+56a^{5}-24a^{4}-28a^{3}+62a^{2}-52a-9$, $3a^{25}+5a^{24}-5a^{22}-3a^{21}+5a^{20}+7a^{19}+a^{18}-6a^{17}-4a^{16}+4a^{15}+9a^{14}+3a^{13}-7a^{12}-7a^{11}+6a^{10}+14a^{9}+4a^{8}-11a^{7}-9a^{6}+8a^{5}+17a^{4}+6a^{3}-12a^{2}-12a-3$, $6a^{24}+4a^{23}-4a^{21}-8a^{20}-2a^{19}+8a^{18}+9a^{17}-7a^{15}-6a^{14}-5a^{13}+3a^{12}+11a^{11}+11a^{10}-7a^{9}-13a^{8}-10a^{7}+3a^{6}+9a^{5}+20a^{4}+6a^{3}-19a^{2}-31a-1$, $10a^{25}-3a^{24}-8a^{23}+6a^{22}+6a^{21}-10a^{20}-6a^{19}+16a^{18}+2a^{17}-17a^{16}+5a^{15}+13a^{14}-12a^{13}-11a^{12}+19a^{11}+10a^{10}-25a^{9}-2a^{8}+24a^{7}-8a^{6}-20a^{5}+17a^{4}+21a^{3}-29a^{2}-19a-1$, $12a^{25}-17a^{24}+20a^{23}-31a^{22}+23a^{21}-17a^{20}+20a^{19}+2a^{18}-15a^{17}+11a^{16}-34a^{15}+41a^{14}-24a^{13}+32a^{12}-21a^{11}-7a^{10}+5a^{9}-27a^{8}+49a^{7}-30a^{6}+47a^{5}-53a^{4}+5a^{3}-9a^{2}+a+9$
| 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: | \( 2848541941084543.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 2848541941084543.5 \cdot 1}{2\cdot\sqrt{400000206565095837929605898429873236273026387935232}}\cr\approx \mathstrut & 1.07840260039094 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 4*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 - 4*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 - 4*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 - 4*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 $S_{26}$ |
| 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 | R | R | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }$ | $22{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $26$ | $26$ | $1$ | $51$ | |||
|
\(3\)
| 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 | $[\ ]^{20}$ | |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(73\)
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.5.0.1 | $x^{5} + 9 x + 68$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 73.6.0.1 | $x^{6} + 45 x^{3} + 23 x^{2} + 48 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 73.7.0.1 | $x^{7} + 10 x + 68$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(1409\)
| $\Q_{1409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(10429777\)
| $\Q_{10429777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(501\!\cdots\!707\)
| $\Q_{50\!\cdots\!07}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |