Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x - 1)
gp: K = bnfinit(y^26 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 3*x - 1)
\( x^{26} - 3x - 1 \)
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: |
\(225763037455656223128623675917042192019113094401\)
\(\medspace = 97\cdot 1027188157\cdot 761641463933293\cdot 2974955834015001526633\)
| 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: | $97^{1/2}1027188157^{1/2}761641463933293^{1/2}2974955834015001526633^{1/2}\approx 4.751452803676537e+23$ | ||
| Ramified primes: |
\(97\), \(1027188157\), \(761641463933293\), \(2974955834015001526633\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{22576\!\cdots\!94401}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $a^{13}-a-1$, $a^{23}-a^{21}+a^{19}-a^{17}+a^{15}+a^{14}-2a^{13}+a^{10}-2a^{9}-a^{8}+a^{7}+2a^{6}-a^{5}-a^{4}+3a^{2}-a-2$, $a^{24}-a^{23}-a^{21}-a^{20}-a^{19}-a^{17}+2a^{16}+2a^{14}+2a^{13}+2a^{12}+a^{10}-2a^{9}-2a^{7}-3a^{6}-a^{5}+2a^{2}+a+1$, $a^{25}+2a^{24}-3a^{23}+a^{22}+a^{21}-4a^{20}+3a^{19}-a^{18}-3a^{17}+3a^{16}-3a^{15}-2a^{14}+4a^{13}-4a^{12}+a^{11}+3a^{10}-5a^{9}+3a^{8}+2a^{7}-4a^{6}+6a^{5}-4a^{3}+7a^{2}-4a-4$, $4a^{25}-a^{24}+2a^{23}+3a^{21}+a^{19}+2a^{18}+2a^{17}+2a^{15}+2a^{14}+2a^{13}+3a^{11}+3a^{10}+3a^{8}+2a^{7}+5a^{6}-2a^{5}+7a^{4}+a^{3}+6a^{2}-2a-2$, $a^{25}-2a^{24}+2a^{23}-a^{22}+a^{19}-2a^{18}+2a^{17}-a^{16}+a^{12}-3a^{11}+4a^{10}-3a^{9}+a^{8}+a^{6}-3a^{5}+4a^{4}-3a^{3}+a^{2}-a$, $a^{25}-a^{24}-a^{23}-a^{22}-3a^{20}-3a^{19}-2a^{18}+2a^{17}+3a^{16}+3a^{15}+a^{14}+2a^{13}+3a^{12}+2a^{11}-2a^{10}-6a^{9}-6a^{8}-3a^{7}+a^{6}-a^{5}-a^{4}+2a^{3}+9a^{2}+8a+3$, $3a^{25}+2a^{24}-a^{23}+a^{22}-3a^{21}+a^{20}-5a^{19}+3a^{18}+a^{17}+2a^{16}+2a^{15}-a^{14}+2a^{13}-8a^{12}+a^{11}-2a^{10}+3a^{9}+a^{8}+4a^{7}+5a^{6}-7a^{5}-7a^{3}+2a^{2}-4a-3$, $4a^{25}-4a^{24}+5a^{23}-5a^{22}+5a^{21}-5a^{20}+4a^{19}-4a^{18}+4a^{17}-2a^{16}+a^{15}-a^{14}-a^{13}+2a^{12}-3a^{11}+6a^{10}-6a^{9}+6a^{8}-9a^{7}+9a^{6}-8a^{5}+9a^{4}-8a^{3}+8a^{2}-10a-4$, $a^{25}+4a^{23}+a^{21}+a^{20}-2a^{19}+2a^{18}-3a^{17}-a^{16}-2a^{15}-5a^{14}+2a^{13}+4a^{11}+5a^{10}+4a^{8}-3a^{7}+a^{5}-6a^{4}-a^{3}-8a^{2}-3a-1$, $20a^{25}-4a^{24}-14a^{23}+6a^{22}+10a^{21}-3a^{20}-15a^{19}+9a^{18}+18a^{17}-24a^{16}-6a^{15}+31a^{14}-14a^{13}-21a^{12}+23a^{11}+8a^{10}-19a^{9}-8a^{8}+22a^{7}+12a^{6}-40a^{5}+3a^{4}+52a^{3}-35a^{2}-37a-6$, $5a^{25}+2a^{24}-7a^{23}+a^{22}+7a^{21}-5a^{20}-6a^{19}+6a^{18}+4a^{17}-8a^{16}-3a^{15}+11a^{14}-a^{13}-11a^{12}+7a^{11}+8a^{10}-10a^{9}-6a^{8}+11a^{7}+2a^{6}-16a^{5}+2a^{4}+16a^{3}-8a^{2}-13a-1$
| 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: | \( 61866206311105.24 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 61866206311105.24 \cdot 1}{2\cdot\sqrt{225763037455656223128623675917042192019113094401}}\cr\approx \mathstrut & 0.985860535029984 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x - 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 - 3*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 - 3*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 - 3*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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.3.0.1}{3} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(97\)
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.3.0.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 97.5.0.1 | $x^{5} + 3 x + 92$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 97.7.0.1 | $x^{7} + 5 x + 92$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 97.7.0.1 | $x^{7} + 5 x + 92$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(1027188157\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(761641463933293\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(297\!\cdots\!633\)
| $\Q_{29\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |