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Properties

Label 15.15.3798978055...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 7^{10}\cdot 41^{12}$
Root discriminant $937.51$
Ramified primes $5, 7, 41$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5^2 : C_3$ (as 15T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41960414299, -400496140345, 871835831985, 782351217820, -1819133611475, -66419271922, 93889842765, 7784845480, -1017450465, -108676650, 3856747, 558830, -4715, -1230, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1230*x^13 - 4715*x^12 + 558830*x^11 + 3856747*x^10 - 108676650*x^9 - 1017450465*x^8 + 7784845480*x^7 + 93889842765*x^6 - 66419271922*x^5 - 1819133611475*x^4 + 782351217820*x^3 + 871835831985*x^2 - 400496140345*x + 41960414299)
 
gp: K = bnfinit(x^15 - 1230*x^13 - 4715*x^12 + 558830*x^11 + 3856747*x^10 - 108676650*x^9 - 1017450465*x^8 + 7784845480*x^7 + 93889842765*x^6 - 66419271922*x^5 - 1819133611475*x^4 + 782351217820*x^3 + 871835831985*x^2 - 400496140345*x + 41960414299, 1)
 

Normalized defining polynomial

\( x^{15} - 1230 x^{13} - 4715 x^{12} + 558830 x^{11} + 3856747 x^{10} - 108676650 x^{9} - 1017450465 x^{8} + 7784845480 x^{7} + 93889842765 x^{6} - 66419271922 x^{5} - 1819133611475 x^{4} + 782351217820 x^{3} + 871835831985 x^{2} - 400496140345 x + 41960414299 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(379897805506290388370204518377780914306640625=5^{24}\cdot 7^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $937.51$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{205} a^{5} - \frac{1}{5}$, $\frac{1}{410} a^{6} - \frac{1}{410} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{410} a^{7} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{410} a^{8} + \frac{2}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{410} a^{9} + \frac{2}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{84050} a^{10} - \frac{1}{1025} a^{5} - \frac{1}{2} a^{3} - \frac{12}{25}$, $\frac{1}{588350} a^{11} - \frac{1}{2870} a^{9} + \frac{1}{1435} a^{8} + \frac{3}{2870} a^{7} - \frac{1}{7175} a^{6} + \frac{3}{1435} a^{5} + \frac{1}{70} a^{4} - \frac{6}{35} a^{3} + \frac{27}{70} a^{2} + \frac{38}{175} a - \frac{31}{70}$, $\frac{1}{3530100} a^{12} + \frac{19}{3530100} a^{10} - \frac{1}{1435} a^{9} + \frac{1}{5740} a^{8} - \frac{6}{7175} a^{7} - \frac{1}{1148} a^{6} + \frac{6}{7175} a^{5} - \frac{103}{420} a^{4} - \frac{13}{70} a^{3} + \frac{107}{700} a^{2} - \frac{5}{14} a - \frac{113}{300}$, $\frac{1}{3530100} a^{13} + \frac{1}{3530100} a^{11} + \frac{3}{588350} a^{10} - \frac{1}{820} a^{9} - \frac{1}{2050} a^{8} + \frac{1}{1148} a^{7} - \frac{17}{14350} a^{6} + \frac{59}{86100} a^{5} - \frac{9}{70} a^{4} + \frac{47}{700} a^{3} + \frac{2}{7} a^{2} - \frac{899}{2100} a + \frac{54}{175}$, $\frac{1}{5156541970098414709408557555402328171202132100} a^{14} - \frac{23801313628134890279730537713638262611}{257827098504920735470427877770116408560106605} a^{13} + \frac{6419075135648344666310783010664934631}{61387404405933508445339970897646763942882525} a^{12} - \frac{1023376879432878643751635341211778344723}{2578270985049207354704278777701164085601066050} a^{11} - \frac{238817764880418606546651057448424715937}{51565419700984147094085575554023281712021321} a^{10} + \frac{9659507675190963676996173952386729323714}{10480776361988647783350726738622618234150675} a^{9} + \frac{1551506978501434625008563468388173711663}{4192310544795459113340290695449047293660270} a^{8} + \frac{4043560940327974124415977067860426283971}{10480776361988647783350726738622618234150675} a^{7} + \frac{56263244582716564520772068664535672324951}{62884658171931886700104360431735709404904050} a^{6} + \frac{25000276844874747604287752025298462750097}{12576931634386377340020872086347141880980810} a^{5} + \frac{14867517783955543739746521076816325364914}{766886075267462032928101956484581822011025} a^{4} + \frac{3524475649673727805001258310694485194341}{102251476702328271057080260864610909601470} a^{3} - \frac{748369414250935316896522627200757683224503}{1533772150534924065856203912969163644022050} a^{2} + \frac{19192412999692231065213433150765411595236}{766886075267462032928101956484581822011025} a + \frac{291801637011852665921849735524731453443101}{613508860213969626342481565187665457608820}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $14$
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:  \( 221873527966000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_3$ (as 15T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 75
The 11 conjugacy class representatives for $C_5^2 : C_3$
Character table for $C_5^2 : C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$41$41.5.4.4$x^{5} + 8856$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.2$x^{5} + 246$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.5$x^{5} - 53136$$5$$1$$4$$C_5$$[\ ]_{5}$