Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 3)
gp: K = bnfinit(y^26 - 3*y + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 3*x + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 3*x + 3)
\( x^{26} - 3x + 3 \)
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: |
\(-4990246961229047120192449009721354684714164966143\)
\(\medspace = -\,3^{25}\cdot 37\cdot 193\cdot 397\cdot 3673\cdot 565614067938252407382930781\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.65\) | 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: | $3^{25/26}37^{1/2}193^{1/2}397^{1/2}3673^{1/2}565614067938252407382930781^{1/2}\approx 6.979362773204392e+18$ | ||
| Ramified primes: |
\(3\), \(37\), \(193\), \(397\), \(3673\), \(565614067938252407382930781\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-17668\!\cdots\!20703}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a-1$, $a^{9}+a-1$, $a^{25}+a^{24}+a^{17}+a^{16}+a^{14}+a^{13}-a^{12}-a^{9}+a^{6}+a^{5}+2a^{3}+a^{2}-2a-2$, $a^{25}+2a^{24}+a^{23}+a^{22}+a^{21}-a^{20}+a^{19}-2a^{18}+a^{17}-a^{16}-a^{15}-3a^{13}+a^{12}-3a^{11}-2a^{9}-2a^{8}-a^{7}-2a^{6}-2a^{4}-a^{3}-3a^{2}-7$, $2a^{24}+2a^{23}-a^{22}-2a^{21}-3a^{20}-2a^{19}+a^{18}+2a^{17}+2a^{16}-2a^{15}-3a^{14}+a^{13}+3a^{12}+4a^{11}-a^{10}-4a^{9}-3a^{8}-3a^{7}+3a^{6}+4a^{5}+3a^{4}-a^{3}-7a^{2}+a+5$, $a^{24}+2a^{21}-a^{20}+a^{19}+2a^{18}-3a^{17}+3a^{16}-a^{15}-3a^{14}+5a^{13}-5a^{12}+a^{11}+3a^{10}-7a^{9}+5a^{8}-2a^{7}-4a^{6}+6a^{5}-4a^{4}+a^{3}+4a^{2}-3a+1$, $a^{25}-2a^{24}+a^{21}+a^{20}-2a^{19}+2a^{18}-2a^{17}+a^{16}-4a^{15}+a^{14}+a^{13}+2a^{11}-a^{10}+7a^{9}-3a^{8}+2a^{7}-4a^{6}+2a^{5}-a^{4}-6a^{3}+3a^{2}-4a+5$, $a^{25}+4a^{24}+4a^{23}+a^{22}-3a^{21}-4a^{20}+5a^{18}+5a^{17}-4a^{15}-4a^{14}+6a^{12}+7a^{11}-7a^{9}-6a^{8}+a^{7}+7a^{6}+7a^{5}+a^{4}-8a^{3}-10a^{2}+a+8$, $a^{25}-a^{24}+2a^{23}+2a^{22}-a^{21}+a^{20}+2a^{19}-2a^{18}-a^{17}+a^{16}-2a^{15}-a^{14}+a^{13}-2a^{12}-a^{11}+a^{10}-3a^{9}+3a^{7}-2a^{6}+6a^{4}-3a^{3}-2a^{2}+7a-5$, $20a^{25}-9a^{24}-27a^{23}-5a^{22}+28a^{21}+22a^{20}-17a^{19}-33a^{18}+38a^{16}+23a^{15}-28a^{14}-40a^{13}+8a^{12}+50a^{11}+22a^{10}-43a^{9}-48a^{8}+21a^{7}+65a^{6}+16a^{5}-63a^{4}-54a^{3}+41a^{2}+82a-58$, $7a^{25}+6a^{24}+5a^{23}+2a^{22}-a^{21}-3a^{20}-5a^{19}-5a^{18}-3a^{17}+4a^{15}+6a^{14}+8a^{13}+7a^{12}+2a^{11}-a^{10}-7a^{9}-8a^{8}-9a^{7}-5a^{6}+3a^{4}+7a^{3}+9a^{2}+5a-17$, $4a^{25}+5a^{24}+24a^{23}+31a^{22}+2a^{21}-51a^{20}-84a^{19}-62a^{18}+5a^{17}+66a^{16}+75a^{15}+33a^{14}-13a^{13}-18a^{12}+15a^{11}+36a^{10}+3a^{9}-70a^{8}-116a^{7}-78a^{6}+28a^{5}+120a^{4}+122a^{3}+37a^{2}-52a-74$
| 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: | \( 62788452030563.16 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 62788452030563.16 \cdot 2}{2\cdot\sqrt{4990246961229047120192449009721354684714164966143}}\cr\approx \mathstrut & 0.668585641439098 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 3)
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 + 3, 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 + 3);
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 + 3);
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $26$ | $21{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | R | $15{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $26$ | $26$ | $1$ | $25$ | |||
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.21.0.1 | $x^{21} - x + 15$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
|
\(193\)
| $\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 193.22.0.1 | $x^{22} + x^{2} - x + 77$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
|
\(397\)
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(3673\)
| $\Q_{3673}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(565\!\cdots\!781\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |