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Properties

Label 16.12.1123091063...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{6}$
Root discriminant $367.83$
Ramified primes $2, 5, 71, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25951599025, 0, -39508786100, 0, 8646624140, 0, 166231270, 0, -32992534, 0, -67600, 0, 25877, 0, -310, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 310*x^14 + 25877*x^12 - 67600*x^10 - 32992534*x^8 + 166231270*x^6 + 8646624140*x^4 - 39508786100*x^2 + 25951599025)
 
gp: K = bnfinit(x^16 - 310*x^14 + 25877*x^12 - 67600*x^10 - 32992534*x^8 + 166231270*x^6 + 8646624140*x^4 - 39508786100*x^2 + 25951599025, 1)
 

Normalized defining polynomial

\( x^{16} - 310 x^{14} + 25877 x^{12} - 67600 x^{10} - 32992534 x^{8} + 166231270 x^{6} + 8646624140 x^{4} - 39508786100 x^{2} + 25951599025 \)

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:  \(112309106399498694797369730826240000000000=2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $367.83$
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:  $2, 5, 71, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{60} a^{12} - \frac{1}{4} a^{11} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{7}{60} a^{8} + \frac{1}{4} a^{6} - \frac{29}{60} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12}$, $\frac{1}{60} a^{13} - \frac{1}{6} a^{11} - \frac{2}{15} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{29}{60} a^{5} - \frac{1}{12} a^{3} + \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{17784224021046206269121323001744279100} a^{14} + \frac{2857552701579531020288852560840157}{355684480420924125382426460034885582} a^{12} - \frac{12395471189792966666023427553286327}{523065412383711949091803617698361150} a^{10} - \frac{1}{4} a^{9} - \frac{294310034898274411780437334971399221}{3556844804209241253824264600348855820} a^{8} - \frac{1}{4} a^{7} - \frac{5541196565514847930563793735469756999}{17784224021046206269121323001744279100} a^{6} + \frac{92692082062383391272938959174321977}{1185614934736413751274754866782951940} a^{4} + \frac{3036552629316573437321434344091253}{11738761730063502487868860067157940} a^{2} + \frac{1}{4} a - \frac{1491416650408328477268010370288467}{3521628519019050746360658020147382}$, $\frac{1}{5673167462713739799849702037556425032900} a^{15} - \frac{646373108701868501160537471608943253}{226926698508549591993988081502257001316} a^{13} + \frac{4483447054031964413038302671467623374}{83428933275202055880142677022888603425} a^{11} - \frac{1}{4} a^{10} - \frac{8595010377243723354729222323601621977}{378211164180915986656646802503761668860} a^{9} + \frac{39466933767227315923881245261407784363}{2836583731356869899924851018778212516450} a^{7} + \frac{1}{4} a^{6} + \frac{246598742681425583633149076371267617493}{567316746271373979984970203755642503290} a^{5} - \frac{21062522806686647946579877939583333}{510636135257762358222295412921370390} a^{3} + \frac{1}{4} a^{2} - \frac{323614579314850812359774511092583029}{748932998378051458726033272284676572} a - \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 527883930962000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{355}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{71}) \), 4.4.203656400.1, 4.4.2525.1, \(\Q(\sqrt{5}, \sqrt{71})\), 8.8.41475929260960000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71Data not computed
101Data not computed