Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*x + 1)
gp: K = bnfinit(y^14 - 3*y^13 - 5*y^12 + 18*y^11 - 7*y^10 + y^9 + 3*y^8 + 71*y^6 - 193*y^5 - 59*y^4 + 34*y^3 + 35*y^2 + 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*x + 1)
\( x^{14} - 3 x^{13} - 5 x^{12} + 18 x^{11} - 7 x^{10} + x^{9} + 3 x^{8} + 71 x^{6} - 193 x^{5} - 59 x^{4} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-8533901005667958784\)
\(\medspace = -\,2^{20}\cdot 19\cdot 809^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.50\) | 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: | $2^{13/6}19^{1/2}809^{1/2}\approx 556.6506474683611$ | ||
| Ramified primes: |
\(2\), \(19\), \(809\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8127659012}a^{13}+\frac{140372866}{2031914753}a^{12}+\frac{626251543}{8127659012}a^{11}+\frac{328948309}{8127659012}a^{10}-\frac{40551404}{2031914753}a^{9}-\frac{1628946855}{8127659012}a^{8}-\frac{32627327}{4063829506}a^{7}-\frac{404881316}{2031914753}a^{6}+\frac{4061419839}{8127659012}a^{5}+\frac{1968457979}{4063829506}a^{4}-\frac{2393012081}{8127659012}a^{3}+\frac{2976307331}{8127659012}a^{2}+\frac{390268958}{2031914753}a+\frac{3944125081}{8127659012}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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: | $10$ | 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: |
$\frac{503805997}{2031914753}a^{13}-\frac{3500143843}{4063829506}a^{12}-\frac{3460083497}{4063829506}a^{11}+\frac{9990832417}{2031914753}a^{10}-\frac{16050342763}{4063829506}a^{9}+\frac{7395580677}{4063829506}a^{8}-\frac{228665564}{2031914753}a^{7}+\frac{238253786}{2031914753}a^{6}+\frac{35932737959}{2031914753}a^{5}-\frac{231196535241}{4063829506}a^{4}+\frac{45860218533}{4063829506}a^{3}+\frac{13458162998}{2031914753}a^{2}+\frac{23285897271}{4063829506}a+\frac{1532701007}{4063829506}$, $\frac{2577005479}{4063829506}a^{13}-\frac{8181560187}{4063829506}a^{12}-\frac{11473482341}{4063829506}a^{11}+\frac{48371669631}{4063829506}a^{10}-\frac{13116593349}{2031914753}a^{9}+\frac{3659524468}{2031914753}a^{8}+\frac{2715854619}{2031914753}a^{7}-\frac{798342697}{4063829506}a^{6}+\frac{184359746565}{4063829506}a^{5}-\frac{265839181955}{2031914753}a^{4}-\frac{61733863117}{4063829506}a^{3}+\frac{50297967043}{2031914753}a^{2}+\frac{38587281331}{2031914753}a+\frac{15396204089}{4063829506}$, $\frac{2788085683}{8127659012}a^{13}-\frac{9852416795}{8127659012}a^{12}-\frac{2189163094}{2031914753}a^{11}+\frac{55516213847}{8127659012}a^{10}-\frac{49617851215}{8127659012}a^{9}+\frac{6351463331}{2031914753}a^{8}-\frac{131802877}{4063829506}a^{7}-\frac{457502059}{2031914753}a^{6}+\frac{200577932987}{8127659012}a^{5}-\frac{642187658359}{8127659012}a^{4}+\frac{88884708779}{4063829506}a^{3}+\frac{54209477845}{8127659012}a^{2}+\frac{37705157617}{8127659012}a+\frac{274983835}{4063829506}$, $\frac{1070682311}{4063829506}a^{13}-\frac{7152732405}{8127659012}a^{12}-\frac{8084340985}{8127659012}a^{11}+\frac{20393743275}{4063829506}a^{10}-\frac{30150870795}{8127659012}a^{9}+\frac{15603795573}{8127659012}a^{8}+\frac{1001791289}{4063829506}a^{7}-\frac{712059433}{4063829506}a^{6}+\frac{76563616373}{4063829506}a^{5}-\frac{463345036905}{8127659012}a^{4}+\frac{41987140219}{8127659012}a^{3}+\frac{5652626652}{2031914753}a^{2}+\frac{33699490459}{8127659012}a+\frac{2330681953}{8127659012}$, $\frac{576168503}{4063829506}a^{13}-\frac{3372887025}{8127659012}a^{12}-\frac{6048289025}{8127659012}a^{11}+\frac{10260136299}{4063829506}a^{10}-\frac{6652312905}{8127659012}a^{9}-\frac{503308345}{8127659012}a^{8}+\frac{2589120001}{4063829506}a^{7}-\frac{50541508}{2031914753}a^{6}+\frac{41250265411}{4063829506}a^{5}-\frac{217203634691}{8127659012}a^{4}-\frac{88139657625}{8127659012}a^{3}+\frac{24513403877}{4063829506}a^{2}+\frac{30099657545}{8127659012}a+\frac{8350915593}{8127659012}$, $\frac{374802767}{738878092}a^{13}-\frac{319822166}{184719523}a^{12}-\frac{1371741963}{738878092}a^{11}+\frac{7396083425}{738878092}a^{10}-\frac{1395516060}{184719523}a^{9}+\frac{2206290705}{738878092}a^{8}+\frac{160342024}{184719523}a^{7}-\frac{211253377}{369439046}a^{6}+\frac{26643144715}{738878092}a^{5}-\frac{20807694703}{184719523}a^{4}+\frac{10044725383}{738878092}a^{3}+\frac{13967726073}{738878092}a^{2}+\frac{2561616263}{369439046}a+\frac{1018561723}{738878092}$, $a$, $\frac{3961189757}{8127659012}a^{13}-\frac{12716804591}{8127659012}a^{12}-\frac{8502524967}{4063829506}a^{11}+\frac{74602217963}{8127659012}a^{10}-\frac{44443714301}{8127659012}a^{9}+\frac{3813294062}{2031914753}a^{8}+\frac{5055788539}{4063829506}a^{7}-\frac{975057357}{2031914753}a^{6}+\frac{283356554337}{8127659012}a^{5}-\frac{822057256671}{8127659012}a^{4}-\frac{13910935903}{2031914753}a^{3}+\frac{127798705109}{8127659012}a^{2}+\frac{81738685851}{8127659012}a+\frac{11360251465}{4063829506}$, $\frac{215035532}{2031914753}a^{13}-\frac{4314346911}{8127659012}a^{12}+\frac{1509543767}{8127659012}a^{11}+\frac{5564066572}{2031914753}a^{10}-\frac{40066638435}{8127659012}a^{9}+\frac{24426545393}{8127659012}a^{8}-\frac{1220082173}{2031914753}a^{7}-\frac{1162206921}{2031914753}a^{6}+\frac{31781959185}{4063829506}a^{5}-\frac{290324591697}{8127659012}a^{4}+\frac{329778082137}{8127659012}a^{3}+\frac{2108587347}{2031914753}a^{2}-\frac{55704459203}{8127659012}a-\frac{10986612753}{8127659012}$, $\frac{115310223}{738878092}a^{13}-\frac{238417495}{369439046}a^{12}-\frac{101134777}{738878092}a^{11}+\frac{2458372099}{738878092}a^{10}-\frac{878822523}{184719523}a^{9}+\frac{2656039873}{738878092}a^{8}-\frac{183006639}{184719523}a^{7}-\frac{124214721}{369439046}a^{6}+\frac{8428298423}{738878092}a^{5}-\frac{15796194993}{369439046}a^{4}+\frac{24391630325}{738878092}a^{3}-\frac{5926209437}{738878092}a^{2}-\frac{1051666345}{369439046}a+\frac{618434511}{738878092}$
| 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: | \( 29284.3445382 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{3}\cdot 29284.3445382 \cdot 1}{2\cdot\sqrt{8533901005667958784}}\cr\approx \mathstrut & 0.318281416485 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*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^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*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^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*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^14 - 3*x^13 - 5*x^12 + 18*x^11 - 7*x^10 + x^9 + 3*x^8 + 71*x^6 - 193*x^5 - 59*x^4 + 34*x^3 + 35*x^2 + 11*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
$C_2^7.\GL(3,2)$ (as 14T51):
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 21504 |
| The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
| Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
| 7.7.670188544.1 |
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
| Degree 14 sibling: | data not computed |
| Degree 28 siblings: | data not computed |
| Degree 42 siblings: | data not computed |
| 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 | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.12.20.37 | $x^{12} + 10 x^{11} + 51 x^{10} + 176 x^{9} + 450 x^{8} + 870 x^{7} + 1299 x^{6} + 1516 x^{5} + 1250 x^{4} + 542 x^{3} + 67 x^{2} - 56 x + 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
|
\(19\)
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(809\)
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |