Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x - 2)
gp: K = bnfinit(y^26 - 3*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 3*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 3*x - 2)
\( x^{26} - 3x - 2 \)
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: |
\(225969602545338033154314948479481790757100825857\)
\(\medspace = 22739663746473685987\cdot 9937244678052017295688440011\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(66.27\) | 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: | $22739663746473685987^{1/2}9937244678052017295688440011^{1/2}\approx 4.7536260112185734e+23$ | ||
| Ramified primes: |
\(22739663746473685987\), \(9937244678052017295688440011\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{22596\!\cdots\!25857}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{25}-2a^{24}+2a^{23}-2a^{22}+a^{21}+a^{20}-2a^{19}+3a^{18}-3a^{17}+a^{16}-2a^{14}+3a^{13}-2a^{12}+2a^{11}-a^{9}+a^{8}-2a^{7}-a^{4}+2a^{3}-1$, $2a^{25}-2a^{24}+a^{23}-a^{22}+a^{17}-a^{16}+2a^{15}-2a^{14}+2a^{13}-2a^{12}+a^{11}-a^{10}+a^{6}+2a^{4}-a^{3}+2a^{2}-2a-5$, $6a^{25}-4a^{24}+2a^{23}-a^{22}+a^{15}-a^{14}+2a^{13}-a^{12}+a^{11}+a^{8}-a^{7}+2a^{6}-2a^{5}+a^{4}-a^{3}-a^{2}-19$, $a^{23}-a^{21}-a^{20}+a^{19}+a^{18}-a^{16}+a^{13}-a^{11}-2a^{10}+a^{9}+3a^{8}+a^{7}-3a^{6}-2a^{5}+a^{4}+2a^{3}-a-1$, $a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+1$, $a^{17}+a^{16}-2a^{13}-a^{12}+2a^{9}+a^{8}-a^{7}-a^{6}-3a^{5}-a^{4}+a^{3}+a^{2}+2a+1$, $5a^{25}-5a^{24}+4a^{23}-3a^{22}+3a^{21}-4a^{20}+3a^{19}-2a^{18}+3a^{17}-3a^{16}+2a^{15}-a^{14}+2a^{13}-2a^{12}+a^{9}-a^{8}-2a^{7}+a^{6}+a^{5}+a^{4}-3a^{3}+2a^{2}-13$, $a^{23}+4a^{22}+a^{21}+a^{19}+a^{18}-4a^{17}-4a^{16}-a^{15}-2a^{14}-4a^{13}+a^{12}+5a^{11}+2a^{10}+2a^{9}+6a^{8}+5a^{7}-3a^{6}-3a^{5}-a^{4}-6a^{3}-11a^{2}-2a+3$, $2a^{25}-a^{23}-2a^{22}-2a^{21}+a^{19}+3a^{18}+2a^{17}+a^{16}-2a^{15}-4a^{14}-4a^{13}-3a^{12}+a^{11}+3a^{10}+5a^{9}+3a^{8}-4a^{6}-6a^{5}-4a^{4}+6a^{2}+9a+3$, $a^{25}-2a^{24}+2a^{23}-a^{22}-2a^{21}+3a^{20}-2a^{18}+3a^{17}-a^{16}-2a^{15}+a^{14}+a^{13}-a^{12}+3a^{10}-3a^{9}-a^{8}+4a^{7}-4a^{6}+4a^{4}-3a^{3}+4a-3$, $a^{24}-a^{23}+a^{21}-a^{20}+a^{18}-a^{17}-a^{16}+2a^{15}-a^{14}-a^{13}+2a^{12}-2a^{10}+a^{9}+2a^{8}-3a^{7}+a^{6}+2a^{5}-3a^{4}+2a^{2}-a-1$, $2a^{25}-3a^{24}+4a^{23}-3a^{22}+2a^{21}-a^{20}+2a^{18}-3a^{17}+2a^{16}+a^{15}-2a^{14}+2a^{13}-a^{12}+2a^{11}-a^{10}-a^{9}+4a^{8}-3a^{7}+a^{6}+a^{5}-a^{4}+2a^{3}-4a^{2}+4a-5$, $a^{25}+a^{23}+a^{22}-a^{20}-2a^{19}-a^{18}+a^{17}+a^{16}+2a^{15}+2a^{14}+a^{13}-a^{12}-4a^{11}-3a^{10}-a^{9}+a^{8}+3a^{7}+3a^{6}+4a^{5}+a^{4}-4a^{3}-5a^{2}-4a-3$
| 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: | \( 71305060830185.08 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 71305060830185.08 \cdot 1}{2\cdot\sqrt{225969602545338033154314948479481790757100825857}}\cr\approx \mathstrut & 1.13575265261185 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 3*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 - 3*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 - 3*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 - 3*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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $26$ | $22{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{3}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/53.11.0.1}{11} }$ | $19{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(22739663746473685987\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(993\!\cdots\!011\)
| $\Q_{99\!\cdots\!11}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |