Properties

Label 25.5.190...049.1
Degree $25$
Signature $[5, 10]$
Discriminant $1.900\times 10^{29}$
Root discriminant $14.83$
Ramified primes $7, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5\times D_5$ (as 25T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 7, -12, -21, 32, 29, -23, 16, -35, -112, 59, 120, -29, -28, 2, -2, 0, -28, 19, 18, -20, 5, 4, -4, 1]);
 

\( x^{25} - 4 x^{24} + 4 x^{23} + 5 x^{22} - 20 x^{21} + 18 x^{20} + 19 x^{19} - 28 x^{18} - 2 x^{16} + 2 x^{15} - 28 x^{14} - 29 x^{13} + 120 x^{12} + 59 x^{11} - 112 x^{10} - 35 x^{9} + 16 x^{8} - 23 x^{7} + 29 x^{6} + 32 x^{5} - 21 x^{4} - 12 x^{3} + 7 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(190035222333323626806494766049\)\(\medspace = 7^{10}\cdot 11^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.83$
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:  $7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1422429784482782} a^{24} + \frac{140614422730571}{1422429784482782} a^{23} - \frac{104578792346784}{711214892241391} a^{22} - \frac{165233546399600}{711214892241391} a^{21} - \frac{144093055169530}{711214892241391} a^{20} - \frac{144067137096092}{711214892241391} a^{19} + \frac{328858474526599}{711214892241391} a^{18} + \frac{11388431288854}{711214892241391} a^{17} + \frac{554089679032639}{1422429784482782} a^{16} - \frac{249580344224563}{711214892241391} a^{15} - \frac{61285048475461}{711214892241391} a^{14} - \frac{195315440452465}{711214892241391} a^{13} + \frac{690630018986355}{1422429784482782} a^{12} + \frac{124898007688931}{711214892241391} a^{11} - \frac{428902554332149}{1422429784482782} a^{10} + \frac{638431435676121}{1422429784482782} a^{9} + \frac{261915975651208}{711214892241391} a^{8} + \frac{169378685340181}{1422429784482782} a^{7} + \frac{47853678613913}{1422429784482782} a^{6} - \frac{256967440247395}{711214892241391} a^{5} - \frac{168316319972885}{1422429784482782} a^{4} - \frac{249305403652881}{711214892241391} a^{3} + \frac{87804121646247}{711214892241391} a^{2} - \frac{188177807611685}{711214892241391} a + \frac{685877223418807}{1422429784482782}$

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:  $14$
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:  \( 40076.42524869914 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 40076.42524869914 \cdot 1}{2\sqrt{190035222333323626806494766049}}\approx 0.141055683863403$ (assuming GRH)

Galois group

$C_5\times D_5$ (as 25T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $C_5\times D_5$
Character table for $C_5\times D_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 5.1.717409.1

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

In the table, R denotes a ramified prime. 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
$7$7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.77.10t1.a.b$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.d$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.77.10t1.a.a$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.77.10t1.a.c$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 2.77.10t6.a.c$2$ $ 7 \cdot 11 $ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.847.10t6.a.a$2$ $ 7 \cdot 11^{2}$ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.847.5t2.a.b$2$ $ 7 \cdot 11^{2}$ 5.1.717409.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.847.10t6.a.c$2$ $ 7 \cdot 11^{2}$ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.847.10t6.a.b$2$ $ 7 \cdot 11^{2}$ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.847.10t6.a.d$2$ $ 7 \cdot 11^{2}$ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.77.10t6.a.b$2$ $ 7 \cdot 11 $ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.77.10t6.a.a$2$ $ 7 \cdot 11 $ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.847.5t2.a.a$2$ $ 7 \cdot 11^{2}$ 5.1.717409.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.77.10t6.a.d$2$ $ 7 \cdot 11 $ 25.5.190035222333323626806494766049.1 $C_5\times D_5$ (as 25T3) $0$ $0$

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 *.