Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 4)
gp: K = bnfinit(y^26 - 3*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 3*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 3*x + 4)
\( x^{26} - 3x + 4 \)
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: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-6930948698829842481802115671756536951850447649657599\)
\(\medspace = -\,2063\cdot 73532749\cdot 551943907461461\cdot 82778530085158225548195457\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(98.60\) | 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: | $2063^{1/2}73532749^{1/2}551943907461461^{1/2}82778530085158225548195457^{1/2}\approx 8.32523194801793e+25$ | ||
| Ramified primes: |
\(2063\), \(73532749\), \(551943907461461\), \(82778530085158225548195457\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-69309\!\cdots\!57599}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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
$C_{2}$, which has order $2$ (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: | $12$ | 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: |
$a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-3a+3$, $2a^{25}+a^{24}-a^{23}-2a^{22}-2a^{21}+a^{20}+a^{19}+2a^{18}-2a^{14}-a^{13}-2a^{12}+3a^{11}+3a^{10}+2a^{9}-2a^{8}-4a^{7}-a^{6}+2a^{4}-a^{3}+3a^{2}+2a-3$, $2a^{25}-2a^{24}+2a^{23}-4a^{22}+2a^{21}-6a^{20}+6a^{19}-11a^{18}+10a^{17}-14a^{16}+12a^{15}-12a^{14}+15a^{13}-15a^{12}+20a^{11}-20a^{10}+23a^{9}-17a^{8}+18a^{7}-14a^{6}+14a^{5}-17a^{4}+15a^{3}-14a^{2}+5a-5$, $6a^{25}+7a^{24}-3a^{23}-9a^{22}-a^{21}+10a^{20}+7a^{19}-8a^{18}-13a^{17}+a^{16}+15a^{15}+7a^{14}-12a^{13}-13a^{12}+6a^{11}+17a^{10}+2a^{9}-18a^{8}-14a^{7}+13a^{6}+26a^{5}+2a^{4}-29a^{3}-21a^{2}+20a+15$, $6a^{25}-11a^{24}+14a^{23}-8a^{22}+4a^{21}+9a^{20}-16a^{19}+18a^{18}-16a^{17}+5a^{16}+12a^{15}-19a^{14}+30a^{13}-23a^{12}+7a^{11}+10a^{10}-31a^{9}+42a^{8}-33a^{7}+17a^{6}+14a^{5}-44a^{4}+56a^{3}-56a^{2}+25a-3$, $4a^{25}+3a^{24}+4a^{23}+2a^{22}+2a^{21}+5a^{20}+6a^{19}+7a^{18}+3a^{17}+5a^{16}+2a^{15}+5a^{14}+7a^{13}+7a^{12}+4a^{11}-a^{10}+5a^{9}+2a^{8}+7a^{7}+4a^{6}+4a^{5}-3a^{4}-5a^{3}+6a^{2}+a-7$, $19a^{25}-6a^{24}-27a^{23}-20a^{22}+10a^{21}+32a^{20}+21a^{19}-14a^{18}-39a^{17}-25a^{16}+19a^{15}+48a^{14}+29a^{13}-22a^{12}-57a^{11}-37a^{10}+26a^{9}+68a^{8}+42a^{7}-28a^{6}-76a^{5}-51a^{4}+32a^{3}+87a^{2}+53a-95$, $15a^{25}+9a^{24}+9a^{23}-8a^{22}+4a^{21}-14a^{20}-14a^{19}-20a^{18}-16a^{17}-38a^{16}-22a^{15}-35a^{14}-42a^{13}-39a^{12}-35a^{11}-53a^{10}-43a^{9}-34a^{8}-62a^{7}-37a^{6}-42a^{5}-43a^{4}-56a^{3}-5a^{2}-69a-59$, $66a^{25}+53a^{24}+35a^{23}+24a^{22}+24a^{21}+28a^{20}+30a^{19}+29a^{18}+21a^{17}+a^{16}-24a^{15}-37a^{14}-32a^{13}-22a^{12}-19a^{11}-22a^{10}-32a^{9}-55a^{8}-83a^{7}-92a^{6}-73a^{5}-49a^{4}-41a^{3}-45a^{2}-55a-275$, $79a^{25}+19a^{24}-79a^{23}-66a^{22}+53a^{21}+107a^{20}+4a^{19}-116a^{18}-71a^{17}+88a^{16}+128a^{15}-26a^{14}-157a^{13}-58a^{12}+144a^{11}+145a^{10}-83a^{9}-214a^{8}-31a^{7}+222a^{6}+164a^{5}-151a^{4}-267a^{3}+18a^{2}+306a-85$, $3a^{25}-5a^{24}-9a^{23}-9a^{22}-11a^{21}-3a^{20}-11a^{19}-9a^{18}-2a^{17}-3a^{16}+9a^{15}+3a^{14}+12a^{13}+15a^{12}+a^{11}+13a^{10}+5a^{9}+14a^{8}+6a^{7}-17a^{6}+2a^{5}-14a^{4}-6a^{3}-9a^{2}-23a-1$, $181a^{25}-165a^{24}+81a^{23}+48a^{22}-178a^{21}+259a^{20}-247a^{19}+137a^{18}+45a^{17}-240a^{16}+372a^{15}-373a^{14}+226a^{13}+36a^{12}-326a^{11}+531a^{10}-554a^{9}+361a^{8}+6a^{7}-428a^{6}+748a^{5}-822a^{4}+578a^{3}-63a^{2}-561a+521$
| 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: | \( 3960421616437175.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 3960421616437175.5 \cdot 2}{2\cdot\sqrt{6930948698829842481802115671756536951850447649657599}}\cr\approx \mathstrut & 1.13157506459725 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 4)
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 + 4, 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 + 4);
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 + 4);
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $26$ | $21{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $26$ | $16{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $26$ | $24{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2063\)
| 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(73532749\)
| $\Q_{73532749}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73532749}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(551943907461461\)
| $\Q_{551943907461461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{551943907461461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{551943907461461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(827\!\cdots\!457\)
| $\Q_{82\!\cdots\!57}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{82\!\cdots\!57}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |