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Properties

Label 16.0.17299727231...1968.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 17^{8}\cdot 4129^{3}\cdot 5897^{3}$
Root discriminant $436.39$
Ramified primes $2, 17, 4129, 5897$
Class number $11776$ (GRH)
Class group $[2, 2, 4, 736]$ (GRH)
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61963913988, 6030300648, 24010475054, -273863064, 9450227107, -3490609372, 2733539310, -621534848, 267942545, -30902744, 9191478, -514872, 121143, -3428, 638, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 638*x^14 - 3428*x^13 + 121143*x^12 - 514872*x^11 + 9191478*x^10 - 30902744*x^9 + 267942545*x^8 - 621534848*x^7 + 2733539310*x^6 - 3490609372*x^5 + 9450227107*x^4 - 273863064*x^3 + 24010475054*x^2 + 6030300648*x + 61963913988)
 
gp: K = bnfinit(x^16 - 8*x^15 + 638*x^14 - 3428*x^13 + 121143*x^12 - 514872*x^11 + 9191478*x^10 - 30902744*x^9 + 267942545*x^8 - 621534848*x^7 + 2733539310*x^6 - 3490609372*x^5 + 9450227107*x^4 - 273863064*x^3 + 24010475054*x^2 + 6030300648*x + 61963913988, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 638 x^{14} - 3428 x^{13} + 121143 x^{12} - 514872 x^{11} + 9191478 x^{10} - 30902744 x^{9} + 267942545 x^{8} - 621534848 x^{7} + 2733539310 x^{6} - 3490609372 x^{5} + 9450227107 x^{4} - 273863064 x^{3} + 24010475054 x^{2} + 6030300648 x + 61963913988 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1729972723111358956551996577891668583251968=2^{34}\cdot 17^{8}\cdot 4129^{3}\cdot 5897^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $436.39$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 17, 4129, 5897$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{12004} a^{14} + \frac{511}{6002} a^{13} + \frac{1811}{12004} a^{12} - \frac{1173}{6002} a^{11} - \frac{1447}{6002} a^{10} + \frac{1497}{3001} a^{9} - \frac{1155}{3001} a^{8} + \frac{113}{3001} a^{7} + \frac{5857}{12004} a^{6} + \frac{2605}{6002} a^{5} - \frac{2245}{12004} a^{4} + \frac{1201}{6002} a^{3} - \frac{1485}{6002} a^{2} - \frac{513}{3001} a + \frac{1285}{3001}$, $\frac{1}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{15} - \frac{63740528533168257703478822996618307235736220004544015039884150953891401897411}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{14} - \frac{306309000441367225171349577728041811786653996811190091040862366723065360750281339}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{13} + \frac{270777308393923336581718747579609849199270983588600897465174482356435196056173255}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{12} + \frac{28213039835672359017681773912473969608338077055491809481448965411617651761165192}{138046001200883431560924437802213273643975344084469811222992503191962999929847727} a^{11} + \frac{17981047532804869856848097265261506593795574499287647672248929629542840430912869}{276092002401766863121848875604426547287950688168939622445985006383925999859695454} a^{10} - \frac{37289089823183491888374026511683051951985484022860497290451257210822280561317711}{138046001200883431560924437802213273643975344084469811222992503191962999929847727} a^{9} - \frac{138215664886170098770189777189296931486757830526162643838543478223721262455517774}{414138003602650294682773313406639820931926032253409433668977509575888999789543181} a^{8} + \frac{197176376477743639155333349250010717250137435317174069106813733827798765664290425}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{7} + \frac{174462739134637055167953473382489294819637251348479124445449794752172133954713677}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{6} + \frac{254506709305973481966590942751845368196283260268936208878554687887371407756740903}{552184004803533726243697751208853094575901376337879244891970012767851999719390908} a^{5} + \frac{6038847958962604267851235274534638917355724709524832613351755320339561293872155}{1656552014410601178731093253626559283727704129013637734675910038303555999158172724} a^{4} - \frac{60407164364125571502251911532283015993285018393314651636551002190286594412593030}{414138003602650294682773313406639820931926032253409433668977509575888999789543181} a^{3} - \frac{36503162686406528826133965021767372530151285730044619246120772197497513271389799}{276092002401766863121848875604426547287950688168939622445985006383925999859695454} a^{2} + \frac{12267486879472677062160909297817943363840863394657363133863021215308027733581884}{414138003602650294682773313406639820931926032253409433668977509575888999789543181} a - \frac{47322801715722207776401653312870409933914923877676435576775814265907731495555117}{138046001200883431560924437802213273643975344084469811222992503191962999929847727}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{736}$, which has order $11776$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 60333969654.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.33318975217221632.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
4129Data not computed
5897Data not computed