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Properties

Label 16.12.3108786751...0961.1
Degree $16$
Signature $[12, 2]$
Discriminant $31^{10}\cdot 41^{14}$
Root discriminant $220.44$
Ramified primes $31, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-74218553, -157790396, 33715784, 174753904, -34040624, -68240968, 28711844, 1964046, -3133015, 632968, 36738, -43896, 5075, 888, -155, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 155*x^14 + 888*x^13 + 5075*x^12 - 43896*x^11 + 36738*x^10 + 632968*x^9 - 3133015*x^8 + 1964046*x^7 + 28711844*x^6 - 68240968*x^5 - 34040624*x^4 + 174753904*x^3 + 33715784*x^2 - 157790396*x - 74218553)
 
gp: K = bnfinit(x^16 - 4*x^15 - 155*x^14 + 888*x^13 + 5075*x^12 - 43896*x^11 + 36738*x^10 + 632968*x^9 - 3133015*x^8 + 1964046*x^7 + 28711844*x^6 - 68240968*x^5 - 34040624*x^4 + 174753904*x^3 + 33715784*x^2 - 157790396*x - 74218553, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 155 x^{14} + 888 x^{13} + 5075 x^{12} - 43896 x^{11} + 36738 x^{10} + 632968 x^{9} - 3133015 x^{8} + 1964046 x^{7} + 28711844 x^{6} - 68240968 x^{5} - 34040624 x^{4} + 174753904 x^{3} + 33715784 x^{2} - 157790396 x - 74218553 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31087867512274251337747808514764310961=31^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.44$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{62} a^{14} + \frac{13}{62} a^{13} - \frac{7}{31} a^{12} + \frac{13}{62} a^{11} - \frac{1}{62} a^{10} - \frac{3}{62} a^{9} + \frac{1}{62} a^{8} + \frac{19}{62} a^{7} - \frac{5}{62} a^{6} - \frac{7}{31} a^{5} - \frac{4}{31} a^{4} + \frac{3}{31} a^{3} + \frac{9}{62} a^{2} - \frac{13}{31} a - \frac{17}{62}$, $\frac{1}{317344167723211931371521157207086990700314275916781334} a^{15} + \frac{2528404937795409566286834344105209803460622458978937}{317344167723211931371521157207086990700314275916781334} a^{14} + \frac{4698328007251415054391204100211140093969558089541184}{158672083861605965685760578603543495350157137958390667} a^{13} - \frac{6374143424590217274686768219818679648452795107487776}{158672083861605965685760578603543495350157137958390667} a^{12} - \frac{2895214610399224309252869315739892392510450820147481}{13797572509704866581370485095960303943491925039860058} a^{11} - \frac{61894250748652230029158324751635216425321466353466631}{317344167723211931371521157207086990700314275916781334} a^{10} + \frac{36215999707588582856308911027944704844512025773941142}{158672083861605965685760578603543495350157137958390667} a^{9} - \frac{12159625495266430476525953231776722408296398153010717}{158672083861605965685760578603543495350157137958390667} a^{8} + \frac{77941084049570228532196926863295277369482605998692123}{158672083861605965685760578603543495350157137958390667} a^{7} + \frac{74774327976444977842475710571845983128719067484375338}{158672083861605965685760578603543495350157137958390667} a^{6} + \frac{18571425417009224790322480601926424764137121327226819}{158672083861605965685760578603543495350157137958390667} a^{5} + \frac{3182326296092480653352835579422301148540430077649523}{317344167723211931371521157207086990700314275916781334} a^{4} - \frac{30693419032808668079568587918313455377866868223187704}{158672083861605965685760578603543495350157137958390667} a^{3} + \frac{11831488646021613374019912942024864176141964612775001}{158672083861605965685760578603543495350157137958390667} a^{2} - \frac{4126409089854402335810048501416369468360523065750453}{317344167723211931371521157207086990700314275916781334} a - \frac{30087557890311076456430446503750912154028071893832862}{158672083861605965685760578603543495350157137958390667}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 17427220933800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
31Data not computed
41Data not computed