Properties

Label 25.1.335...881.1
Degree $25$
Signature $[1, 12]$
Discriminant $3.352\times 10^{41}$
Root discriminant $45.82$
Ramified prime $2887$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125)
 
gp: K = bnfinit(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2125, -3850, -2655, 17869, 35311, -1959, -65478, -41862, 21048, 18271, 11956, 13530, -6966, -11301, 8388, -3146, -2567, 4254, -2994, 1425, -519, 184, -53, 18, -5, 1]);
 

\( x^{25} - 5 x^{24} + 18 x^{23} - 53 x^{22} + 184 x^{21} - 519 x^{20} + 1425 x^{19} - 2994 x^{18} + 4254 x^{17} - 2567 x^{16} - 3146 x^{15} + 8388 x^{14} - 11301 x^{13} - 6966 x^{12} + 13530 x^{11} + 11956 x^{10} + 18271 x^{9} + 21048 x^{8} - 41862 x^{7} - 65478 x^{6} - 1959 x^{5} + 35311 x^{4} + 17869 x^{3} - 2655 x^{2} - 3850 x - 2125 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(335244303153752999188341104839710238574881\)\(\medspace = 2887^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.82$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2887$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{2}{9} a^{6} + \frac{2}{9} a^{4} + \frac{2}{9} a^{2} + \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{135} a^{17} + \frac{1}{135} a^{16} - \frac{4}{135} a^{15} - \frac{4}{135} a^{14} - \frac{1}{27} a^{13} - \frac{1}{27} a^{12} + \frac{1}{45} a^{11} + \frac{2}{45} a^{10} - \frac{1}{15} a^{9} - \frac{1}{9} a^{8} + \frac{2}{15} a^{7} - \frac{13}{45} a^{6} + \frac{2}{27} a^{5} - \frac{11}{135} a^{4} - \frac{43}{135} a^{3} + \frac{1}{27} a^{2} - \frac{14}{135} a + \frac{5}{27}$, $\frac{1}{135} a^{18} - \frac{1}{27} a^{16} - \frac{1}{135} a^{14} + \frac{8}{135} a^{12} + \frac{1}{45} a^{11} - \frac{1}{9} a^{10} - \frac{2}{45} a^{9} - \frac{4}{45} a^{8} - \frac{4}{45} a^{7} - \frac{41}{135} a^{6} + \frac{8}{45} a^{5} + \frac{13}{135} a^{4} - \frac{14}{45} a^{3} + \frac{26}{135} a^{2} - \frac{17}{45} a + \frac{13}{27}$, $\frac{1}{135} a^{19} + \frac{1}{27} a^{16} - \frac{2}{45} a^{15} - \frac{1}{27} a^{14} - \frac{2}{135} a^{13} - \frac{7}{135} a^{12} + \frac{1}{9} a^{11} - \frac{2}{45} a^{10} + \frac{1}{45} a^{9} + \frac{2}{15} a^{8} - \frac{11}{135} a^{7} + \frac{13}{45} a^{6} + \frac{1}{45} a^{5} - \frac{22}{135} a^{4} + \frac{7}{45} a^{3} - \frac{41}{135} a^{2} + \frac{5}{27} a - \frac{5}{27}$, $\frac{1}{135} a^{20} + \frac{4}{135} a^{16} + \frac{1}{45} a^{14} + \frac{1}{45} a^{13} - \frac{4}{27} a^{12} + \frac{1}{15} a^{11} + \frac{1}{45} a^{10} + \frac{1}{45} a^{9} + \frac{19}{135} a^{8} + \frac{1}{15} a^{7} - \frac{4}{45} a^{6} - \frac{4}{45} a^{5} + \frac{46}{135} a^{4} + \frac{2}{5} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{10}{27}$, $\frac{1}{1215} a^{21} - \frac{4}{1215} a^{20} - \frac{1}{405} a^{19} - \frac{1}{1215} a^{18} - \frac{2}{81} a^{16} + \frac{52}{1215} a^{15} - \frac{22}{1215} a^{14} - \frac{2}{405} a^{13} + \frac{107}{1215} a^{12} + \frac{8}{135} a^{11} - \frac{11}{81} a^{10} + \frac{29}{243} a^{9} - \frac{94}{1215} a^{8} + \frac{2}{81} a^{7} + \frac{56}{1215} a^{6} + \frac{47}{405} a^{5} - \frac{131}{405} a^{4} + \frac{116}{243} a^{3} - \frac{313}{1215} a^{2} - \frac{67}{135} a - \frac{44}{243}$, $\frac{1}{1215} a^{22} - \frac{1}{1215} a^{20} - \frac{4}{1215} a^{19} - \frac{4}{1215} a^{18} - \frac{1}{405} a^{17} - \frac{59}{1215} a^{16} + \frac{8}{405} a^{15} - \frac{58}{1215} a^{14} - \frac{16}{1215} a^{13} + \frac{77}{1215} a^{12} + \frac{32}{405} a^{11} + \frac{187}{1215} a^{10} - \frac{1}{15} a^{9} + \frac{158}{1215} a^{8} - \frac{17}{243} a^{7} + \frac{122}{1215} a^{6} - \frac{62}{135} a^{5} + \frac{16}{1215} a^{4} + \frac{43}{135} a^{3} - \frac{334}{1215} a^{2} - \frac{98}{243} a + \frac{85}{243}$, $\frac{1}{21130065} a^{23} - \frac{13}{128061} a^{22} + \frac{19}{112995} a^{21} - \frac{61}{414315} a^{20} + \frac{13988}{21130065} a^{19} - \frac{33641}{21130065} a^{18} + \frac{20182}{21130065} a^{17} - \frac{261223}{7043355} a^{16} + \frac{208807}{21130065} a^{15} + \frac{2572}{2347785} a^{14} + \frac{67712}{21130065} a^{13} - \frac{1348631}{21130065} a^{12} + \frac{738277}{21130065} a^{11} - \frac{83026}{782595} a^{10} + \frac{1531012}{21130065} a^{9} + \frac{148085}{1408671} a^{8} + \frac{265186}{4226013} a^{7} + \frac{6728329}{21130065} a^{6} + \frac{2724778}{21130065} a^{5} + \frac{279161}{640305} a^{4} + \frac{6831256}{21130065} a^{3} - \frac{205718}{1408671} a^{2} + \frac{4871678}{21130065} a - \frac{74219}{248589}$, $\frac{1}{496530265982080463839989245611729575} a^{24} + \frac{419225209025069077673173433}{33102017732138697589332616374115305} a^{23} + \frac{11032906276111054246631806456253}{45139115089280042167271749601066325} a^{22} - \frac{1751739820871622333005138118658}{9735887568276087526274298933563325} a^{21} + \frac{1008842758606142805286504460046449}{496530265982080463839989245611729575} a^{20} - \frac{1478199503512858379904216265545434}{496530265982080463839989245611729575} a^{19} - \frac{59802709639841288755672148433691}{99306053196416092767997849122345915} a^{18} + \frac{438781793709696718418688146743297}{165510088660693487946663081870576525} a^{17} - \frac{3728168491257120155107801759234736}{496530265982080463839989245611729575} a^{16} + \frac{2412559152950704364281800478338637}{55170029553564495982221027290192175} a^{15} - \frac{6987249014475691363022342980767796}{496530265982080463839989245611729575} a^{14} + \frac{19866306579284203662298792767593173}{496530265982080463839989245611729575} a^{13} + \frac{50971763844417339261658460817562234}{496530265982080463839989245611729575} a^{12} - \frac{6102275638924902491028990197574052}{165510088660693487946663081870576525} a^{11} + \frac{14787980792366976380522185778762678}{99306053196416092767997849122345915} a^{10} + \frac{5675673859505806834519944222640147}{165510088660693487946663081870576525} a^{9} - \frac{51392334939215603849811316576660909}{496530265982080463839989245611729575} a^{8} + \frac{78485829262118052581697320713900948}{496530265982080463839989245611729575} a^{7} + \frac{7995020567397633284878215759410818}{496530265982080463839989245611729575} a^{6} - \frac{1836787002467496023354297618861162}{55170029553564495982221027290192175} a^{5} - \frac{211548423345947369513262491841635984}{496530265982080463839989245611729575} a^{4} - \frac{20643387278441579442336497275211518}{165510088660693487946663081870576525} a^{3} + \frac{6315298023530577724083400796054774}{16017105354260660123870620826184825} a^{2} - \frac{1541310560270983132434627693367642}{3203421070852132024774124165236965} a - \frac{142696284227162555181451220591771}{389435502731043501050971957342533}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 305458559273.9288 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 305458559273.9288 \cdot 1}{2\sqrt{335244303153752999188341104839710238574881}}\approx 1.99723889942240$ (assuming GRH)

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.8334769.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $25$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ $25$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2887Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2887.2t1.a.a$1$ $ 2887 $ \(\Q(\sqrt{-2887}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2887.5t2.a.a$2$ $ 2887 $ 5.1.8334769.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2887.5t2.a.b$2$ $ 2887 $ 5.1.8334769.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2887.25t4.a.f$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.d$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.h$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.c$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.i$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.g$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.a$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.j$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.e$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.b$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.