Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712)
gp: K = bnfinit(y^12 - 3*y^11 - 5*y^10 + 11*y^9 + 40*y^8 + 139*y^7 - 1505*y^6 + 9090*y^5 - 24504*y^4 + 5576*y^3 + 95056*y^2 - 153440*y + 59712, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712)
\( x^{12} - 3 x^{11} - 5 x^{10} + 11 x^{9} + 40 x^{8} + 139 x^{7} - 1505 x^{6} + 9090 x^{5} - 24504 x^{4} + \cdots + 59712 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(36442794820015380778081\)
\(\medspace = 661^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(75.88\) | 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: | $661^{4/5}\approx 180.36869686739456$ | ||
| Ramified primes: |
\(661\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{3}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{16}a^{5}+\frac{5}{16}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{91\!\cdots\!48}a^{11}-\frac{15\!\cdots\!55}{91\!\cdots\!48}a^{10}-\frac{27\!\cdots\!17}{91\!\cdots\!48}a^{9}-\frac{73\!\cdots\!61}{91\!\cdots\!48}a^{8}+\frac{38\!\cdots\!53}{57\!\cdots\!78}a^{7}-\frac{73\!\cdots\!45}{15\!\cdots\!68}a^{6}+\frac{10\!\cdots\!27}{91\!\cdots\!48}a^{5}-\frac{13\!\cdots\!27}{45\!\cdots\!24}a^{4}+\frac{10\!\cdots\!11}{28\!\cdots\!39}a^{3}+\frac{46\!\cdots\!41}{11\!\cdots\!56}a^{2}-\frac{10\!\cdots\!69}{57\!\cdots\!78}a+\frac{20\!\cdots\!87}{28\!\cdots\!39}$
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: | $7$ | 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: |
$\frac{92\!\cdots\!59}{45\!\cdots\!24}a^{11}-\frac{21\!\cdots\!73}{45\!\cdots\!24}a^{10}-\frac{14\!\cdots\!19}{45\!\cdots\!24}a^{9}+\frac{14\!\cdots\!93}{45\!\cdots\!24}a^{8}+\frac{23\!\cdots\!07}{57\!\cdots\!78}a^{7}+\frac{32\!\cdots\!13}{75\!\cdots\!84}a^{6}-\frac{29\!\cdots\!55}{45\!\cdots\!24}a^{5}+\frac{18\!\cdots\!37}{22\!\cdots\!12}a^{4}+\frac{13\!\cdots\!91}{11\!\cdots\!56}a^{3}-\frac{36\!\cdots\!17}{11\!\cdots\!56}a^{2}+\frac{87\!\cdots\!43}{28\!\cdots\!39}a-\frac{61\!\cdots\!46}{28\!\cdots\!39}$, $\frac{13\!\cdots\!45}{45\!\cdots\!24}a^{11}-\frac{33\!\cdots\!49}{45\!\cdots\!24}a^{10}-\frac{12\!\cdots\!55}{45\!\cdots\!24}a^{9}+\frac{64\!\cdots\!81}{45\!\cdots\!24}a^{8}+\frac{43\!\cdots\!53}{22\!\cdots\!12}a^{7}+\frac{47\!\cdots\!07}{75\!\cdots\!84}a^{6}-\frac{19\!\cdots\!91}{45\!\cdots\!24}a^{5}+\frac{65\!\cdots\!85}{28\!\cdots\!39}a^{4}-\frac{61\!\cdots\!25}{11\!\cdots\!56}a^{3}-\frac{71\!\cdots\!83}{11\!\cdots\!56}a^{2}+\frac{14\!\cdots\!07}{57\!\cdots\!78}a-\frac{33\!\cdots\!59}{28\!\cdots\!39}$, $\frac{12\!\cdots\!69}{28\!\cdots\!39}a^{11}+\frac{99\!\cdots\!97}{45\!\cdots\!24}a^{10}-\frac{15\!\cdots\!05}{45\!\cdots\!24}a^{9}-\frac{43\!\cdots\!99}{45\!\cdots\!24}a^{8}+\frac{67\!\cdots\!17}{45\!\cdots\!24}a^{7}+\frac{41\!\cdots\!47}{37\!\cdots\!92}a^{6}-\frac{19\!\cdots\!13}{45\!\cdots\!24}a^{5}+\frac{13\!\cdots\!85}{45\!\cdots\!24}a^{4}-\frac{10\!\cdots\!83}{57\!\cdots\!78}a^{3}-\frac{13\!\cdots\!13}{11\!\cdots\!56}a^{2}+\frac{72\!\cdots\!83}{57\!\cdots\!78}a-\frac{50\!\cdots\!72}{28\!\cdots\!39}$, $\frac{25\!\cdots\!77}{45\!\cdots\!24}a^{11}-\frac{19\!\cdots\!01}{22\!\cdots\!12}a^{10}-\frac{97\!\cdots\!75}{22\!\cdots\!12}a^{9}-\frac{87\!\cdots\!49}{57\!\cdots\!78}a^{8}+\frac{11\!\cdots\!81}{45\!\cdots\!24}a^{7}+\frac{79\!\cdots\!25}{75\!\cdots\!84}a^{6}-\frac{13\!\cdots\!75}{22\!\cdots\!12}a^{5}+\frac{18\!\cdots\!99}{45\!\cdots\!24}a^{4}-\frac{21\!\cdots\!30}{28\!\cdots\!39}a^{3}-\frac{10\!\cdots\!15}{11\!\cdots\!56}a^{2}+\frac{20\!\cdots\!79}{57\!\cdots\!78}a-\frac{69\!\cdots\!18}{28\!\cdots\!39}$, $\frac{37\!\cdots\!27}{91\!\cdots\!48}a^{11}+\frac{37\!\cdots\!89}{91\!\cdots\!48}a^{10}-\frac{16\!\cdots\!21}{91\!\cdots\!48}a^{9}-\frac{54\!\cdots\!33}{91\!\cdots\!48}a^{8}-\frac{33\!\cdots\!47}{45\!\cdots\!24}a^{7}+\frac{59\!\cdots\!69}{15\!\cdots\!68}a^{6}-\frac{39\!\cdots\!25}{91\!\cdots\!48}a^{5}+\frac{21\!\cdots\!85}{11\!\cdots\!56}a^{4}-\frac{12\!\cdots\!69}{11\!\cdots\!56}a^{3}-\frac{20\!\cdots\!73}{28\!\cdots\!39}a^{2}+\frac{59\!\cdots\!35}{57\!\cdots\!78}a-\frac{28\!\cdots\!25}{28\!\cdots\!39}$, $\frac{64\!\cdots\!65}{91\!\cdots\!48}a^{11}-\frac{78\!\cdots\!15}{91\!\cdots\!48}a^{10}-\frac{45\!\cdots\!81}{91\!\cdots\!48}a^{9}-\frac{10\!\cdots\!17}{91\!\cdots\!48}a^{8}+\frac{74\!\cdots\!16}{28\!\cdots\!39}a^{7}+\frac{21\!\cdots\!11}{15\!\cdots\!68}a^{6}-\frac{73\!\cdots\!37}{91\!\cdots\!48}a^{5}+\frac{22\!\cdots\!57}{45\!\cdots\!24}a^{4}-\frac{96\!\cdots\!03}{11\!\cdots\!56}a^{3}-\frac{62\!\cdots\!93}{57\!\cdots\!78}a^{2}+\frac{13\!\cdots\!33}{28\!\cdots\!39}a-\frac{68\!\cdots\!41}{28\!\cdots\!39}$, $\frac{57\!\cdots\!31}{57\!\cdots\!78}a^{11}+\frac{32\!\cdots\!69}{22\!\cdots\!12}a^{10}-\frac{44\!\cdots\!63}{22\!\cdots\!12}a^{9}-\frac{69\!\cdots\!71}{22\!\cdots\!12}a^{8}+\frac{30\!\cdots\!97}{22\!\cdots\!12}a^{7}+\frac{13\!\cdots\!33}{18\!\cdots\!96}a^{6}-\frac{35\!\cdots\!95}{22\!\cdots\!12}a^{5}-\frac{74\!\cdots\!13}{22\!\cdots\!12}a^{4}+\frac{84\!\cdots\!59}{57\!\cdots\!78}a^{3}-\frac{43\!\cdots\!67}{11\!\cdots\!56}a^{2}+\frac{23\!\cdots\!79}{28\!\cdots\!39}a-\frac{21\!\cdots\!24}{28\!\cdots\!39}$
| 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: | \( 31968632.5563 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 31968632.5563 \cdot 1}{2\cdot\sqrt{36442794820015380778081}}\cr\approx \mathstrut & 2.08798648086 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712)
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^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712, 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^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712);
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^12 - 3*x^11 - 5*x^10 + 11*x^9 + 40*x^8 + 139*x^7 - 1505*x^6 + 9090*x^5 - 24504*x^4 + 5576*x^3 + 95056*x^2 - 153440*x + 59712);
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 7920 |
| The 10 conjugacy class representatives for $M_{11}$ |
| Character table for $M_{11}$ |
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]
Sibling fields
| Degree 11 sibling: | data not computed |
| Degree 22 sibling: | data not computed |
| Minimal sibling: | 11.3.36442794820015380778081.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(661\)
| $\Q_{661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $5$ | $5$ | $1$ | $4$ | ||||
| Deg $5$ | $5$ | $1$ | $4$ |