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Properties

Label 15.1.145...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-1.452\times 10^{32}$
Root discriminant \(139.36\)
Ramified primes $2,5,7,11$
Class number $15$ (GRH)
Class group [15] (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264)
 
Copy content gp:K = bnfinit(y^15 - 3*y^14 + 56*y^13 - 40*y^12 + 1313*y^11 + 1645*y^10 + 23526*y^9 + 60814*y^8 + 448956*y^7 + 1131964*y^6 + 5118440*y^5 + 10233032*y^4 + 27247040*y^3 + 38644224*y^2 + 50848768*y + 46155264, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264)
 

\( x^{15} - 3 x^{14} + 56 x^{13} - 40 x^{12} + 1313 x^{11} + 1645 x^{10} + 23526 x^{9} + 60814 x^{8} + \cdots + 46155264 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $15$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(1, 7)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-145248802764051296834191360000000\) \(\medspace = -\,2^{21}\cdot 5^{7}\cdot 7^{10}\cdot 11^{12}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(139.36\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{1/2}7^{2/3}11^{4/5}\approx 157.59514608543674$
Ramified primes:   \(2\), \(5\), \(7\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-10}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{20}a^{7}+\frac{1}{10}a^{6}+\frac{1}{5}a^{5}-\frac{1}{10}a^{4}+\frac{9}{20}a^{3}+\frac{2}{5}a^{2}+\frac{3}{10}a-\frac{2}{5}$, $\frac{1}{120}a^{8}-\frac{1}{60}a^{7}-\frac{7}{60}a^{6}-\frac{1}{15}a^{5}+\frac{7}{120}a^{4}-\frac{19}{60}a^{3}+\frac{17}{60}a^{2}+\frac{7}{30}a-\frac{2}{5}$, $\frac{1}{600}a^{9}-\frac{1}{600}a^{8}-\frac{1}{60}a^{7}+\frac{1}{12}a^{6}-\frac{37}{600}a^{5}+\frac{77}{600}a^{4}-\frac{5}{12}a^{3}+\frac{1}{12}a^{2}+\frac{32}{75}a+\frac{11}{25}$, $\frac{1}{600}a^{10}-\frac{1}{600}a^{8}-\frac{1}{60}a^{7}-\frac{37}{600}a^{6}-\frac{1}{60}a^{5}+\frac{107}{600}a^{4}+\frac{1}{3}a^{3}-\frac{97}{300}a^{2}-\frac{7}{15}a+\frac{1}{25}$, $\frac{1}{14400}a^{11}-\frac{7}{14400}a^{10}+\frac{1}{2400}a^{9}+\frac{1}{1440}a^{8}-\frac{347}{14400}a^{7}-\frac{1571}{14400}a^{6}+\frac{307}{3600}a^{5}-\frac{4}{45}a^{4}-\frac{19}{400}a^{3}+\frac{1487}{3600}a^{2}-\frac{37}{450}a+\frac{11}{30}$, $\frac{1}{22032000}a^{12}+\frac{61}{7344000}a^{11}-\frac{581}{1101600}a^{10}-\frac{8569}{11016000}a^{9}-\frac{5279}{1296000}a^{8}+\frac{163619}{22032000}a^{7}+\frac{49669}{440640}a^{6}+\frac{281797}{2754000}a^{5}-\frac{684383}{5508000}a^{4}-\frac{1369123}{5508000}a^{3}+\frac{7631}{36720}a^{2}+\frac{47261}{688500}a-\frac{52693}{114750}$, $\frac{1}{3965760000}a^{13}-\frac{11}{3965760000}a^{12}-\frac{18971}{1982880000}a^{11}+\frac{203027}{660960000}a^{10}-\frac{261937}{1321920000}a^{9}+\frac{14067001}{3965760000}a^{8}+\frac{94159}{15491250}a^{7}+\frac{10098023}{123930000}a^{6}-\frac{5143633}{330480000}a^{5}-\frac{17778749}{110160000}a^{4}-\frac{2943967}{247860000}a^{3}+\frac{2027863}{7290000}a^{2}+\frac{11194547}{30982500}a+\frac{677089}{2581875}$, $\frac{1}{49\cdots 00}a^{14}-\frac{107958437}{24\cdots 00}a^{13}+\frac{21077491537}{16\cdots 00}a^{12}-\frac{1103958146741}{15\cdots 00}a^{11}+\frac{131799177163087}{54\cdots 00}a^{10}-\frac{13\cdots 93}{24\cdots 00}a^{9}+\frac{65\cdots 61}{49\cdots 00}a^{8}-\frac{18\cdots 19}{12\cdots 00}a^{7}-\frac{12\cdots 91}{12\cdots 00}a^{6}+\frac{44\cdots 39}{51\cdots 00}a^{5}-\frac{59\cdots 93}{24\cdots 00}a^{4}+\frac{202336728846413}{630771556230000}a^{3}-\frac{306590679364451}{18\cdots 00}a^{2}+\frac{18\cdots 93}{95\cdots 25}a+\frac{815975916771883}{31\cdots 75}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$, $5$

Class group and class number

Ideal class group:  $C_{15}$, which has order $15$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{15}$, which has order $15$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{305010143801}{12\cdots 00}a^{14}-\frac{1017459084917}{12\cdots 00}a^{13}+\frac{4973730322253}{40\cdots 00}a^{12}-\frac{14577715561331}{12\cdots 00}a^{11}+\frac{1988871777739}{80\cdots 00}a^{10}+\frac{164196660154637}{12\cdots 00}a^{9}+\frac{52\cdots 93}{12\cdots 00}a^{8}+\frac{65\cdots 27}{12\cdots 00}a^{7}+\frac{57\cdots 43}{76\cdots 00}a^{6}+\frac{99\cdots 31}{10\cdots 00}a^{5}+\frac{12\cdots 73}{30\cdots 00}a^{4}-\frac{10\cdots 31}{34\cdots 00}a^{3}-\frac{11\cdots 19}{19\cdots 50}a^{2}-\frac{29\cdots 21}{38\cdots 00}a-\frac{34\cdots 51}{31\cdots 75}$, $\frac{100192569641}{61\cdots 00}a^{14}+\frac{33975370547}{30\cdots 00}a^{13}+\frac{584445078577}{10\cdots 00}a^{12}+\frac{1755704206711}{24\cdots 00}a^{11}+\frac{16094586803999}{68\cdots 00}a^{10}+\frac{120472853004641}{61\cdots 00}a^{9}+\frac{24257008997051}{30\cdots 00}a^{8}+\frac{51\cdots 51}{12\cdots 00}a^{7}+\frac{10\cdots 59}{61\cdots 00}a^{6}+\frac{167799892695013}{204369984218520}a^{5}+\frac{20\cdots 13}{76\cdots 00}a^{4}+\frac{27\cdots 41}{34\cdots 00}a^{3}+\frac{12\cdots 01}{76\cdots 00}a^{2}+\frac{54\cdots 82}{19\cdots 25}a+\frac{66\cdots 62}{31\cdots 75}$, $\frac{12248376899911}{49\cdots 00}a^{14}-\frac{177190291529959}{49\cdots 00}a^{13}+\frac{87909941742823}{40\cdots 00}a^{12}-\frac{37\cdots 31}{24\cdots 00}a^{11}+\frac{18\cdots 47}{54\cdots 00}a^{10}-\frac{12\cdots 31}{49\cdots 00}a^{9}-\frac{21\cdots 77}{24\cdots 00}a^{8}-\frac{58\cdots 43}{15\cdots 00}a^{7}-\frac{81\cdots 81}{12\cdots 00}a^{6}-\frac{33\cdots 03}{40\cdots 00}a^{5}-\frac{42\cdots 29}{24\cdots 00}a^{4}-\frac{35\cdots 31}{42\cdots 00}a^{3}-\frac{12\cdots 39}{76\cdots 00}a^{2}-\frac{59\cdots 29}{19\cdots 50}a-\frac{12\cdots 52}{31\cdots 75}$, $\frac{59\cdots 23}{24\cdots 00}a^{14}-\frac{30\cdots 93}{61\cdots 00}a^{13}+\frac{62\cdots 41}{81\cdots 00}a^{12}+\frac{10\cdots 91}{61\cdots 00}a^{11}+\frac{35\cdots 41}{27\cdots 00}a^{10}+\frac{28\cdots 23}{61\cdots 00}a^{9}+\frac{80\cdots 33}{24\cdots 00}a^{8}+\frac{73\cdots 93}{61\cdots 00}a^{7}+\frac{33\cdots 27}{61\cdots 00}a^{6}+\frac{16\cdots 43}{10\cdots 00}a^{5}+\frac{10\cdots 03}{24\cdots 00}a^{4}+\frac{72\cdots 53}{85\cdots 00}a^{3}+\frac{37\cdots 01}{30\cdots 00}a^{2}+\frac{27\cdots 91}{19\cdots 50}a+\frac{27\cdots 33}{31\cdots 75}$, $\frac{68\cdots 37}{54\cdots 00}a^{14}-\frac{45\cdots 83}{21\cdots 00}a^{13}+\frac{92\cdots 39}{90\cdots 00}a^{12}-\frac{48\cdots 53}{54\cdots 00}a^{11}+\frac{97\cdots 17}{67\cdots 00}a^{10}-\frac{15\cdots 67}{10\cdots 00}a^{9}+\frac{93\cdots 47}{68\cdots 00}a^{8}-\frac{35\cdots 29}{27\cdots 00}a^{7}+\frac{96\cdots 21}{54\cdots 00}a^{6}-\frac{29\cdots 71}{18\cdots 00}a^{5}+\frac{42\cdots 63}{34\cdots 00}a^{4}-\frac{28\cdots 23}{22\cdots 00}a^{3}-\frac{55\cdots 93}{42\cdots 00}a^{2}-\frac{29\cdots 69}{851541600910500}a-\frac{20\cdots 17}{354809000379375}$, $\frac{87\cdots 57}{16\cdots 00}a^{14}-\frac{64\cdots 83}{16\cdots 00}a^{13}+\frac{11\cdots 97}{27\cdots 00}a^{12}-\frac{14\cdots 67}{81\cdots 00}a^{11}+\frac{21\cdots 89}{18\cdots 00}a^{10}-\frac{25\cdots 71}{96\cdots 00}a^{9}+\frac{59\cdots 73}{40\cdots 00}a^{8}-\frac{40\cdots 87}{12\cdots 00}a^{7}+\frac{50\cdots 53}{40\cdots 00}a^{6}+\frac{29\cdots 49}{13\cdots 00}a^{5}+\frac{19\cdots 69}{25\cdots 00}a^{4}+\frac{17\cdots 56}{10\cdots 25}a^{3}+\frac{15\cdots 23}{638656200682875}a^{2}+\frac{11\cdots 71}{31\cdots 75}a+\frac{23\cdots 31}{10\cdots 25}$, $\frac{34\cdots 07}{49\cdots 00}a^{14}-\frac{11\cdots 09}{49\cdots 00}a^{13}+\frac{29\cdots 53}{81\cdots 00}a^{12}-\frac{65\cdots 71}{24\cdots 00}a^{11}+\frac{38\cdots 71}{54\cdots 00}a^{10}+\frac{58\cdots 79}{49\cdots 00}a^{9}+\frac{71\cdots 27}{61\cdots 00}a^{8}+\frac{42\cdots 41}{12\cdots 00}a^{7}+\frac{29\cdots 19}{12\cdots 00}a^{6}+\frac{22\cdots 07}{40\cdots 00}a^{5}+\frac{66\cdots 39}{30\cdots 00}a^{4}+\frac{48\cdots 59}{11\cdots 00}a^{3}+\frac{55\cdots 97}{76\cdots 00}a^{2}+\frac{38\cdots 37}{38\cdots 00}a+\frac{16\cdots 92}{31\cdots 75}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1496162404931.6238 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1496162404931.6238 \cdot 15}{2\cdot\sqrt{145248802764051296834191360000000}}\cr\approx \mathstrut & 719.901029502197 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 3*x^14 + 56*x^13 - 40*x^12 + 1313*x^11 + 1645*x^10 + 23526*x^9 + 60814*x^8 + 448956*x^7 + 1131964*x^6 + 5118440*x^5 + 10233032*x^4 + 27247040*x^3 + 38644224*x^2 + 50848768*x + 46155264); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{15}$ (as 15T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.1960.1, 5.1.23425600.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 30
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ R R R $15$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/59.5.0.1}{5} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(7\) Copy content Toggle raw display 7.5.3.10a1.3$x^{15} + 3 x^{11} + 12 x^{10} + 3 x^{7} + 24 x^{6} + 48 x^{5} + x^{3} + 12 x^{2} + 48 x + 71$$3$$5$$10$$C_{15}$$$[\ ]_{3}^{5}$$
\(11\) Copy content Toggle raw display 11.3.5.12a1.3$x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 360 x^{10} + 890 x^{9} + 1080 x^{8} + 4940 x^{7} + 8730 x^{6} + 9752 x^{5} + 29880 x^{4} + 39285 x^{3} + 29259 x^{2} + 65632 x + 59049$$5$$3$$12$$C_{15}$$$[\ ]_{5}^{3}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*30 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.40.2t1.b.a$1$ $ 2^{3} \cdot 5 $ \(\Q(\sqrt{-10}) \) $C_2$ (as 2T1) $1$ $-1$
*30 2.1960.3t2.a.a$2$ $ 2^{3} \cdot 5 \cdot 7^{2}$ 3.1.1960.1 $S_3$ (as 3T2) $1$ $0$
*30 2.4840.5t2.a.b$2$ $ 2^{3} \cdot 5 \cdot 11^{2}$ 5.1.23425600.1 $D_{5}$ (as 5T2) $1$ $0$
*30 2.4840.5t2.a.a$2$ $ 2^{3} \cdot 5 \cdot 11^{2}$ 5.1.23425600.1 $D_{5}$ (as 5T2) $1$ $0$
*30 2.237160.15t2.a.d$2$ $ 2^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ 15.1.145248802764051296834191360000000.1 $D_{15}$ (as 15T2) $1$ $0$
*30 2.237160.15t2.a.b$2$ $ 2^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ 15.1.145248802764051296834191360000000.1 $D_{15}$ (as 15T2) $1$ $0$
*30 2.237160.15t2.a.c$2$ $ 2^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ 15.1.145248802764051296834191360000000.1 $D_{15}$ (as 15T2) $1$ $0$
*30 2.237160.15t2.a.a$2$ $ 2^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ 15.1.145248802764051296834191360000000.1 $D_{15}$ (as 15T2) $1$ $0$

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)