Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 4*x - 1)
gp: K = bnfinit(y^26 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 4*x - 1)
\( x^{26} - 4x - 1 \)
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: |
\(5960464477539154233330193268616658399616009\)
\(\medspace = 29\cdot 10128571112394569\cdot 20292423834064993115883509\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.18\) | 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: | $29^{1/2}10128571112394569^{1/2}20292423834064993115883509^{1/2}\approx 2.4414062500000186e+21$ | ||
| Ramified primes: |
\(29\), \(10128571112394569\), \(20292423834064993115883509\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{59604\!\cdots\!16009}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{12}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: |
$a$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{12}-2$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a-1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{17}-\frac{1}{2}a^{12}+\frac{1}{2}a^{4}+a^{2}-2$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{20}-a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+a^{13}+\frac{1}{2}a^{12}-a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{7}+\frac{3}{2}a^{5}-a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a-1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-a^{18}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{3}{2}a^{8}+\frac{1}{2}a^{7}+a^{6}+a^{5}-a^{4}-\frac{3}{2}a^{3}-\frac{1}{2}a^{2}+\frac{5}{2}a$, $\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}+a^{11}+\frac{1}{2}a^{10}+\frac{3}{2}a^{9}+\frac{1}{2}a^{8}+\frac{3}{2}a^{7}+\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-a^{3}-\frac{1}{2}a^{2}-a+1$, $a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{16}-a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}+a^{10}-2a^{9}+2a^{8}-\frac{3}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{3}{2}a^{3}+2a^{2}-\frac{5}{2}a-\frac{5}{2}$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+a^{10}-a^{8}+a^{6}+\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{3}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+a^{21}-a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+a^{8}-a^{7}+\frac{1}{2}a^{6}-a^{5}+2a^{4}-\frac{5}{2}a^{3}+\frac{3}{2}a^{2}-\frac{3}{2}a$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{23}-a^{21}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-a^{16}-a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{3}{2}a^{10}+a^{8}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+a^{3}-a^{2}-\frac{3}{2}a-\frac{3}{2}$, $a^{25}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{17}+\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+a^{6}-a^{5}+\frac{1}{2}a^{4}-a^{3}+\frac{3}{2}a^{2}+a-\frac{3}{2}$
| 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: | \( 168756466576.10596 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 168756466576.10596 \cdot 1}{2\cdot\sqrt{5960464477539154233330193268616658399616009}}\cr\approx \mathstrut & 0.523369954898118 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 4*x - 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 - 4*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 - 4*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 - 4*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 | $24{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $19{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $25{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.10.0.1 | $x^{10} + x^{6} + 25 x^{5} + 8 x^{4} + 17 x^{3} + 2 x^{2} + 22 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 29.12.0.1 | $x^{12} + 3 x^{8} + 19 x^{7} + 28 x^{6} + 9 x^{5} + 16 x^{4} + 25 x^{3} + x^{2} + x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(10128571112394569\)
| $\Q_{10128571112394569}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
|
\(202\!\cdots\!509\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |