Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071)
gp: K = bnfinit(y^15 - 19*y^12 + 36*y^11 - 39*y^10 + 59*y^9 - 126*y^8 + 648*y^7 - 556*y^6 + 585*y^5 + 93*y^4 + 319*y^3 - 297*y^2 + 1116*y + 1071, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071)
\( x^{15} - 19 x^{12} + 36 x^{11} - 39 x^{10} + 59 x^{9} - 126 x^{8} + 648 x^{7} - 556 x^{6} + 585 x^{5} + \cdots + 1071 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-66959890291162926105759\)
\(\medspace = -\,3^{20}\cdot 79^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.24\) | 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: | $3^{4/3}79^{1/2}\approx 38.456983737546345$ | ||
| Ramified primes: |
\(3\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-79}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{27}a^{12}-\frac{2}{9}a^{10}+\frac{4}{27}a^{9}-\frac{1}{3}a^{8}-\frac{4}{9}a^{7}-\frac{11}{27}a^{6}+\frac{1}{3}a^{5}+\frac{2}{9}a^{4}+\frac{1}{27}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{27}a^{13}+\frac{1}{9}a^{11}+\frac{4}{27}a^{10}-\frac{1}{3}a^{9}-\frac{1}{9}a^{8}-\frac{11}{27}a^{7}+\frac{1}{3}a^{6}-\frac{4}{9}a^{5}+\frac{1}{27}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{24\!\cdots\!63}a^{14}-\frac{46\!\cdots\!98}{82\!\cdots\!21}a^{13}+\frac{34\!\cdots\!44}{24\!\cdots\!63}a^{12}+\frac{15\!\cdots\!24}{24\!\cdots\!63}a^{11}-\frac{37\!\cdots\!61}{82\!\cdots\!21}a^{10}+\frac{36\!\cdots\!20}{35\!\cdots\!09}a^{9}+\frac{81\!\cdots\!75}{24\!\cdots\!63}a^{8}+\frac{12\!\cdots\!42}{27\!\cdots\!07}a^{7}-\frac{14\!\cdots\!55}{35\!\cdots\!09}a^{6}-\frac{75\!\cdots\!72}{24\!\cdots\!63}a^{5}+\frac{22\!\cdots\!74}{91\!\cdots\!69}a^{4}-\frac{10\!\cdots\!93}{24\!\cdots\!63}a^{3}-\frac{89\!\cdots\!11}{13\!\cdots\!67}a^{2}+\frac{71\!\cdots\!89}{27\!\cdots\!07}a+\frac{18\!\cdots\!16}{39\!\cdots\!01}$
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
$C_{2}$, which has order $2$
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: |
$\frac{12\!\cdots\!67}{24\!\cdots\!63}a^{14}+\frac{28\!\cdots\!88}{82\!\cdots\!21}a^{13}+\frac{10\!\cdots\!60}{91\!\cdots\!69}a^{12}-\frac{23\!\cdots\!19}{24\!\cdots\!63}a^{11}+\frac{86\!\cdots\!03}{82\!\cdots\!21}a^{10}-\frac{34\!\cdots\!71}{11\!\cdots\!03}a^{9}+\frac{13\!\cdots\!01}{24\!\cdots\!63}a^{8}-\frac{49\!\cdots\!97}{82\!\cdots\!21}a^{7}+\frac{37\!\cdots\!89}{11\!\cdots\!03}a^{6}-\frac{52\!\cdots\!13}{24\!\cdots\!63}a^{5}+\frac{64\!\cdots\!49}{82\!\cdots\!21}a^{4}-\frac{14\!\cdots\!60}{82\!\cdots\!21}a^{3}+\frac{17\!\cdots\!36}{39\!\cdots\!01}a^{2}+\frac{29\!\cdots\!71}{91\!\cdots\!69}a+\frac{74\!\cdots\!52}{13\!\cdots\!67}$, $\frac{68\!\cdots\!87}{24\!\cdots\!63}a^{14}-\frac{38\!\cdots\!01}{82\!\cdots\!21}a^{13}+\frac{40\!\cdots\!53}{91\!\cdots\!69}a^{12}-\frac{11\!\cdots\!08}{24\!\cdots\!63}a^{11}+\frac{15\!\cdots\!69}{82\!\cdots\!21}a^{10}-\frac{41\!\cdots\!47}{11\!\cdots\!03}a^{9}+\frac{93\!\cdots\!63}{24\!\cdots\!63}a^{8}-\frac{39\!\cdots\!44}{82\!\cdots\!21}a^{7}+\frac{26\!\cdots\!43}{11\!\cdots\!03}a^{6}-\frac{11\!\cdots\!05}{24\!\cdots\!63}a^{5}+\frac{49\!\cdots\!05}{82\!\cdots\!21}a^{4}-\frac{99\!\cdots\!14}{82\!\cdots\!21}a^{3}-\frac{37\!\cdots\!84}{39\!\cdots\!01}a^{2}+\frac{58\!\cdots\!01}{91\!\cdots\!69}a+\frac{88\!\cdots\!86}{13\!\cdots\!67}$, $\frac{24\!\cdots\!51}{82\!\cdots\!21}a^{14}-\frac{11\!\cdots\!79}{24\!\cdots\!63}a^{13}-\frac{28\!\cdots\!05}{24\!\cdots\!63}a^{12}-\frac{16\!\cdots\!42}{27\!\cdots\!07}a^{11}+\frac{59\!\cdots\!55}{24\!\cdots\!63}a^{10}-\frac{14\!\cdots\!98}{35\!\cdots\!09}a^{9}-\frac{37\!\cdots\!98}{82\!\cdots\!21}a^{8}-\frac{31\!\cdots\!93}{24\!\cdots\!63}a^{7}+\frac{10\!\cdots\!87}{35\!\cdots\!09}a^{6}-\frac{24\!\cdots\!43}{82\!\cdots\!21}a^{5}+\frac{19\!\cdots\!62}{24\!\cdots\!63}a^{4}-\frac{24\!\cdots\!17}{24\!\cdots\!63}a^{3}+\frac{19\!\cdots\!50}{39\!\cdots\!01}a^{2}-\frac{69\!\cdots\!49}{91\!\cdots\!69}a+\frac{81\!\cdots\!97}{39\!\cdots\!01}$, $\frac{86\!\cdots\!33}{82\!\cdots\!21}a^{14}+\frac{50\!\cdots\!00}{27\!\cdots\!07}a^{13}-\frac{56\!\cdots\!60}{24\!\cdots\!63}a^{12}-\frac{20\!\cdots\!39}{82\!\cdots\!21}a^{11}+\frac{20\!\cdots\!14}{82\!\cdots\!21}a^{10}+\frac{14\!\cdots\!72}{35\!\cdots\!09}a^{9}+\frac{77\!\cdots\!13}{82\!\cdots\!21}a^{8}-\frac{17\!\cdots\!96}{82\!\cdots\!21}a^{7}+\frac{14\!\cdots\!79}{35\!\cdots\!09}a^{6}+\frac{44\!\cdots\!77}{82\!\cdots\!21}a^{5}+\frac{16\!\cdots\!34}{82\!\cdots\!21}a^{4}-\frac{44\!\cdots\!70}{24\!\cdots\!63}a^{3}+\frac{20\!\cdots\!33}{13\!\cdots\!67}a^{2}+\frac{50\!\cdots\!08}{27\!\cdots\!07}a+\frac{54\!\cdots\!75}{39\!\cdots\!01}$, $\frac{66\!\cdots\!31}{24\!\cdots\!63}a^{14}+\frac{11\!\cdots\!48}{24\!\cdots\!63}a^{13}+\frac{54\!\cdots\!13}{24\!\cdots\!63}a^{12}-\frac{12\!\cdots\!49}{24\!\cdots\!63}a^{11}+\frac{38\!\cdots\!42}{24\!\cdots\!63}a^{10}+\frac{12\!\cdots\!29}{35\!\cdots\!09}a^{9}+\frac{47\!\cdots\!28}{24\!\cdots\!63}a^{8}-\frac{69\!\cdots\!12}{24\!\cdots\!63}a^{7}+\frac{44\!\cdots\!37}{35\!\cdots\!09}a^{6}+\frac{39\!\cdots\!91}{24\!\cdots\!63}a^{5}+\frac{14\!\cdots\!78}{24\!\cdots\!63}a^{4}+\frac{59\!\cdots\!79}{24\!\cdots\!63}a^{3}+\frac{67\!\cdots\!56}{13\!\cdots\!67}a^{2}+\frac{11\!\cdots\!13}{27\!\cdots\!07}a+\frac{44\!\cdots\!26}{39\!\cdots\!01}$, $\frac{55\!\cdots\!50}{24\!\cdots\!63}a^{14}+\frac{45\!\cdots\!99}{24\!\cdots\!63}a^{13}-\frac{10\!\cdots\!95}{27\!\cdots\!07}a^{12}-\frac{12\!\cdots\!09}{24\!\cdots\!63}a^{11}+\frac{11\!\cdots\!70}{24\!\cdots\!63}a^{10}+\frac{80\!\cdots\!21}{11\!\cdots\!03}a^{9}+\frac{18\!\cdots\!93}{24\!\cdots\!63}a^{8}-\frac{84\!\cdots\!72}{24\!\cdots\!63}a^{7}+\frac{12\!\cdots\!68}{11\!\cdots\!03}a^{6}+\frac{80\!\cdots\!16}{24\!\cdots\!63}a^{5}-\frac{30\!\cdots\!78}{24\!\cdots\!63}a^{4}-\frac{10\!\cdots\!83}{82\!\cdots\!21}a^{3}+\frac{55\!\cdots\!71}{39\!\cdots\!01}a^{2}+\frac{53\!\cdots\!02}{27\!\cdots\!07}a+\frac{56\!\cdots\!07}{13\!\cdots\!67}$, $\frac{18\!\cdots\!10}{24\!\cdots\!63}a^{14}+\frac{79\!\cdots\!88}{24\!\cdots\!63}a^{13}-\frac{91\!\cdots\!68}{24\!\cdots\!63}a^{12}-\frac{24\!\cdots\!44}{24\!\cdots\!63}a^{11}-\frac{15\!\cdots\!25}{24\!\cdots\!63}a^{10}+\frac{74\!\cdots\!06}{35\!\cdots\!09}a^{9}-\frac{10\!\cdots\!51}{24\!\cdots\!63}a^{8}+\frac{19\!\cdots\!81}{24\!\cdots\!63}a^{7}-\frac{36\!\cdots\!72}{35\!\cdots\!09}a^{6}+\frac{45\!\cdots\!40}{24\!\cdots\!63}a^{5}-\frac{65\!\cdots\!78}{24\!\cdots\!63}a^{4}+\frac{39\!\cdots\!51}{24\!\cdots\!63}a^{3}-\frac{70\!\cdots\!33}{13\!\cdots\!67}a^{2}+\frac{15\!\cdots\!80}{91\!\cdots\!69}a-\frac{14\!\cdots\!45}{39\!\cdots\!01}$
| 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: | \( 145411.427122 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 145411.427122 \cdot 2}{2\cdot\sqrt{66959890291162926105759}}\cr\approx \mathstrut & 0.434490459056 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071)
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^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071, 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^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071);
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^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071);
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 solvable group of order 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.6399.1, 5.1.6241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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
| Galois closure: | deg 30 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15$ | R | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
| 3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
| 3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |