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Properties

Label 14.0.52275321316...1875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 1823^{7}$
Root discriminant $95.47$
Ramified primes $5, 1823$
Class number $1682$ (GRH)
Class group $[29, 58]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63264400, -34197120, 26472884, -5253552, 2094349, 14466, 129647, -11976, 15353, -1614, 817, -96, 47, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 47*x^12 - 96*x^11 + 817*x^10 - 1614*x^9 + 15353*x^8 - 11976*x^7 + 129647*x^6 + 14466*x^5 + 2094349*x^4 - 5253552*x^3 + 26472884*x^2 - 34197120*x + 63264400)
 
gp: K = bnfinit(x^14 - 6*x^13 + 47*x^12 - 96*x^11 + 817*x^10 - 1614*x^9 + 15353*x^8 - 11976*x^7 + 129647*x^6 + 14466*x^5 + 2094349*x^4 - 5253552*x^3 + 26472884*x^2 - 34197120*x + 63264400, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 47 x^{12} - 96 x^{11} + 817 x^{10} - 1614 x^{9} + 15353 x^{8} - 11976 x^{7} + 129647 x^{6} + 14466 x^{5} + 2094349 x^{4} - 5253552 x^{3} + 26472884 x^{2} - 34197120 x + 63264400 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5227532131604246464597421875=-\,5^{7}\cdot 1823^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.47$
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:  $5, 1823$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{7} - \frac{1}{20} a^{6} - \frac{3}{20} a^{5} + \frac{3}{20} a^{4} + \frac{1}{20} a^{3} - \frac{1}{20} a^{2}$, $\frac{1}{60} a^{8} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{9}{20} a^{2} - \frac{2}{5} a + \frac{1}{3}$, $\frac{1}{120} a^{9} - \frac{1}{40} a^{7} - \frac{1}{40} a^{6} + \frac{1}{40} a^{5} + \frac{3}{40} a^{4} - \frac{1}{20} a^{3} + \frac{19}{40} a^{2} - \frac{29}{60} a - \frac{1}{2}$, $\frac{1}{120} a^{10} - \frac{1}{120} a^{8} - \frac{1}{40} a^{7} + \frac{1}{40} a^{6} + \frac{7}{40} a^{5} + \frac{3}{20} a^{4} - \frac{13}{40} a^{3} + \frac{1}{15} a^{2} + \frac{1}{10} a + \frac{1}{3}$, $\frac{1}{23400} a^{11} - \frac{49}{23400} a^{10} + \frac{11}{7800} a^{9} + \frac{43}{11700} a^{8} - \frac{17}{3900} a^{7} + \frac{187}{3900} a^{6} + \frac{487}{7800} a^{5} + \frac{233}{1560} a^{4} + \frac{1849}{4680} a^{3} + \frac{577}{5850} a^{2} + \frac{7}{39} a - \frac{4}{117}$, $\frac{1}{6715800} a^{12} - \frac{1}{129150} a^{11} + \frac{71}{223860} a^{10} + \frac{1529}{516600} a^{9} + \frac{1327}{223860} a^{8} - \frac{22577}{1119300} a^{7} + \frac{2333}{63960} a^{6} - \frac{105643}{1119300} a^{5} - \frac{29663}{335790} a^{4} - \frac{34787}{6715800} a^{3} - \frac{45337}{559650} a^{2} - \frac{33563}{167895} a - \frac{2791}{11193}$, $\frac{1}{4693059175886148702726000} a^{13} + \frac{340643451203717}{27934876046941361325750} a^{12} + \frac{2321593676554643389}{223479008375530890606000} a^{11} + \frac{742423682342083486397}{180502275995621103951000} a^{10} + \frac{1274698604796986816567}{361004551991242207902000} a^{9} - \frac{7876218032225047632821}{2346529587943074351363000} a^{8} + \frac{21580511059894016743613}{1564353058628716234242000} a^{7} + \frac{14049717402599912516851}{391088264657179058560500} a^{6} - \frac{496018662437769194763829}{4693059175886148702726000} a^{5} + \frac{74860521621400609505543}{782176529314358117121000} a^{4} - \frac{1359637635918468311417}{7630990529896176752400} a^{3} + \frac{60664139164944404151019}{2346529587943074351363000} a^{2} + \frac{104142609326666056317593}{234652958794307435136300} a + \frac{1725526993123950012187}{11732647939715371756815}$

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

Class group and class number

$C_{29}\times C_{58}$, which has order $1682$ (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:  $6$
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:  \( 2195415.12282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-9115}) \), 7.1.757303595875.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.757303595875.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
1823Data not computed