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Properties

Label 15.1.704...875.1
Degree $15$
Signature $[1, 7]$
Discriminant $-7.041\times 10^{32}$
Root discriminant \(154.82\)
Ramified primes $3,5,7,11$
Class number $15$ (GRH)
Class group [15] (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980)
 
gp: K = bnfinit(y^15 + 9*y^13 - 137*y^12 + 171*y^11 - 729*y^10 + 6821*y^9 - 18729*y^8 + 68841*y^7 + 73741*y^6 + 1162125*y^5 - 113439*y^4 + 6042802*y^3 - 2166687*y^2 + 1882230*y + 1784980, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980)
 

\( x^{15} + 9 x^{13} - 137 x^{12} + 171 x^{11} - 729 x^{10} + 6821 x^{9} - 18729 x^{8} + 68841 x^{7} + \cdots + 1784980 \) Copy content Toggle raw display

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-704066412430281882891194976796875\) \(\medspace = -\,3^{20}\cdot 5^{7}\cdot 7^{7}\cdot 11^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(154.82\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}5^{1/2}7^{1/2}11^{4/5}\approx 174.30499923069038$
Ramified primes:   \(3\), \(5\), \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-35}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{14}a^{7}+\frac{3}{14}a^{6}-\frac{3}{14}a^{5}+\frac{1}{7}a^{4}+\frac{5}{14}a^{3}+\frac{3}{14}a^{2}-\frac{3}{14}a+\frac{3}{7}$, $\frac{1}{56}a^{8}-\frac{3}{14}a^{6}-\frac{5}{28}a^{5}+\frac{3}{28}a^{4}+\frac{2}{7}a^{3}+\frac{9}{56}a^{2}-\frac{3}{28}a-\frac{1}{14}$, $\frac{1}{280}a^{9}-\frac{1}{140}a^{8}+\frac{1}{70}a^{7}+\frac{31}{140}a^{6}+\frac{17}{140}a^{5}-\frac{6}{35}a^{4}+\frac{1}{280}a^{3}-\frac{4}{35}a^{2}+\frac{5}{14}a-\frac{3}{7}$, $\frac{1}{560}a^{10}-\frac{1}{560}a^{9}-\frac{3}{560}a^{8}+\frac{3}{280}a^{7}+\frac{29}{140}a^{6}-\frac{4}{35}a^{5}-\frac{57}{560}a^{4}-\frac{271}{560}a^{3}+\frac{123}{560}a^{2}-\frac{23}{56}a+\frac{5}{28}$, $\frac{1}{5600}a^{11}-\frac{1}{2800}a^{10}+\frac{1}{700}a^{9}+\frac{9}{5600}a^{8}+\frac{19}{560}a^{7}-\frac{1}{7}a^{6}+\frac{587}{5600}a^{5}+\frac{453}{2800}a^{4}-\frac{47}{200}a^{3}-\frac{933}{5600}a^{2}+\frac{193}{560}a+\frac{111}{280}$, $\frac{1}{196000}a^{12}+\frac{9}{196000}a^{11}+\frac{73}{98000}a^{10}+\frac{257}{196000}a^{9}+\frac{67}{28000}a^{8}+\frac{73}{19600}a^{7}+\frac{34187}{196000}a^{6}+\frac{603}{196000}a^{5}-\frac{3603}{19600}a^{4}+\frac{65951}{196000}a^{3}+\frac{63207}{196000}a^{2}+\frac{199}{560}a+\frac{1801}{9800}$, $\frac{1}{784000}a^{13}-\frac{1}{784000}a^{12}-\frac{1}{16000}a^{11}-\frac{293}{784000}a^{10}-\frac{141}{784000}a^{9}-\frac{1}{156800}a^{8}-\frac{409}{112000}a^{7}+\frac{82933}{784000}a^{6}+\frac{2501}{156800}a^{5}+\frac{7603}{112000}a^{4}+\frac{125677}{784000}a^{3}-\frac{30871}{156800}a^{2}-\frac{38013}{78400}a-\frac{1873}{7840}$, $\frac{1}{66\!\cdots\!00}a^{14}-\frac{79\!\cdots\!89}{16\!\cdots\!00}a^{13}-\frac{47\!\cdots\!91}{33\!\cdots\!00}a^{12}+\frac{33\!\cdots\!39}{94\!\cdots\!00}a^{11}+\frac{32\!\cdots\!93}{33\!\cdots\!00}a^{10}-\frac{35\!\cdots\!03}{33\!\cdots\!00}a^{9}-\frac{16\!\cdots\!17}{47\!\cdots\!00}a^{8}+\frac{99\!\cdots\!29}{33\!\cdots\!00}a^{7}-\frac{54\!\cdots\!69}{47\!\cdots\!00}a^{6}-\frac{21\!\cdots\!89}{33\!\cdots\!00}a^{5}+\frac{75\!\cdots\!91}{33\!\cdots\!00}a^{4}+\frac{11\!\cdots\!93}{47\!\cdots\!00}a^{3}+\frac{79\!\cdots\!19}{66\!\cdots\!00}a^{2}-\frac{25\!\cdots\!03}{26\!\cdots\!92}a+\frac{10\!\cdots\!07}{33\!\cdots\!00}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $5$, $7$

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{90\!\cdots\!03}{16\!\cdots\!00}a^{14}-\frac{26\!\cdots\!37}{13\!\cdots\!00}a^{13}+\frac{31\!\cdots\!81}{66\!\cdots\!00}a^{12}-\frac{20\!\cdots\!27}{27\!\cdots\!00}a^{11}+\frac{74\!\cdots\!93}{66\!\cdots\!00}a^{10}-\frac{48\!\cdots\!23}{13\!\cdots\!00}a^{9}+\frac{35\!\cdots\!39}{94\!\cdots\!00}a^{8}-\frac{28\!\cdots\!17}{26\!\cdots\!20}a^{7}+\frac{35\!\cdots\!29}{94\!\cdots\!00}a^{6}+\frac{44\!\cdots\!99}{13\!\cdots\!00}a^{5}+\frac{40\!\cdots\!71}{66\!\cdots\!00}a^{4}-\frac{46\!\cdots\!47}{18\!\cdots\!00}a^{3}+\frac{18\!\cdots\!47}{66\!\cdots\!00}a^{2}-\frac{13\!\cdots\!91}{66\!\cdots\!00}a+\frac{58\!\cdots\!01}{33\!\cdots\!00}$, $\frac{154752192209799}{24\!\cdots\!00}a^{14}-\frac{17\!\cdots\!03}{24\!\cdots\!00}a^{13}+\frac{49\!\cdots\!93}{12\!\cdots\!00}a^{12}-\frac{23\!\cdots\!49}{17\!\cdots\!00}a^{11}+\frac{13\!\cdots\!33}{12\!\cdots\!00}a^{10}-\frac{14\!\cdots\!79}{12\!\cdots\!00}a^{9}+\frac{11\!\cdots\!41}{17\!\cdots\!00}a^{8}-\frac{14\!\cdots\!53}{24\!\cdots\!00}a^{7}+\frac{25\!\cdots\!93}{17\!\cdots\!00}a^{6}-\frac{34\!\cdots\!01}{12\!\cdots\!00}a^{5}-\frac{86\!\cdots\!17}{24\!\cdots\!00}a^{4}-\frac{12\!\cdots\!71}{17\!\cdots\!00}a^{3}+\frac{18\!\cdots\!83}{61\!\cdots\!00}a^{2}-\frac{28\!\cdots\!29}{12\!\cdots\!40}a-\frac{36\!\cdots\!03}{15\!\cdots\!00}$, $\frac{15\!\cdots\!97}{66\!\cdots\!00}a^{14}-\frac{92\!\cdots\!33}{66\!\cdots\!00}a^{13}+\frac{11\!\cdots\!27}{66\!\cdots\!00}a^{12}-\frac{31\!\cdots\!83}{94\!\cdots\!00}a^{11}+\frac{72\!\cdots\!87}{13\!\cdots\!00}a^{10}-\frac{91\!\cdots\!61}{66\!\cdots\!00}a^{9}+\frac{12\!\cdots\!11}{75\!\cdots\!12}a^{8}-\frac{33\!\cdots\!11}{66\!\cdots\!00}a^{7}+\frac{64\!\cdots\!23}{37\!\cdots\!60}a^{6}+\frac{92\!\cdots\!49}{66\!\cdots\!00}a^{5}+\frac{16\!\cdots\!73}{66\!\cdots\!00}a^{4}-\frac{18\!\cdots\!49}{94\!\cdots\!00}a^{3}+\frac{37\!\cdots\!19}{33\!\cdots\!00}a^{2}-\frac{45\!\cdots\!21}{33\!\cdots\!00}a-\frac{87\!\cdots\!48}{10\!\cdots\!25}$, $\frac{31\!\cdots\!01}{66\!\cdots\!48}a^{14}+\frac{28\!\cdots\!01}{33\!\cdots\!00}a^{13}+\frac{75\!\cdots\!39}{33\!\cdots\!00}a^{12}-\frac{29\!\cdots\!07}{47\!\cdots\!00}a^{11}-\frac{15\!\cdots\!33}{33\!\cdots\!00}a^{10}+\frac{18\!\cdots\!59}{33\!\cdots\!00}a^{9}+\frac{28\!\cdots\!09}{94\!\cdots\!00}a^{8}-\frac{11\!\cdots\!63}{33\!\cdots\!00}a^{7}+\frac{19\!\cdots\!59}{47\!\cdots\!00}a^{6}+\frac{65\!\cdots\!01}{66\!\cdots\!00}a^{5}+\frac{20\!\cdots\!61}{33\!\cdots\!00}a^{4}+\frac{22\!\cdots\!11}{47\!\cdots\!00}a^{3}-\frac{72\!\cdots\!19}{66\!\cdots\!00}a^{2}+\frac{14\!\cdots\!07}{33\!\cdots\!00}a+\frac{76\!\cdots\!43}{33\!\cdots\!40}$, $\frac{46\!\cdots\!51}{33\!\cdots\!00}a^{14}-\frac{36\!\cdots\!53}{66\!\cdots\!00}a^{13}+\frac{44\!\cdots\!77}{66\!\cdots\!00}a^{12}-\frac{32\!\cdots\!59}{13\!\cdots\!00}a^{11}+\frac{12\!\cdots\!41}{13\!\cdots\!00}a^{10}-\frac{77\!\cdots\!11}{66\!\cdots\!00}a^{9}+\frac{24\!\cdots\!83}{18\!\cdots\!00}a^{8}-\frac{39\!\cdots\!01}{66\!\cdots\!00}a^{7}+\frac{30\!\cdots\!49}{18\!\cdots\!00}a^{6}-\frac{11\!\cdots\!81}{66\!\cdots\!00}a^{5}+\frac{57\!\cdots\!43}{66\!\cdots\!00}a^{4}-\frac{67\!\cdots\!99}{94\!\cdots\!00}a^{3}+\frac{15\!\cdots\!73}{66\!\cdots\!00}a^{2}-\frac{24\!\cdots\!97}{66\!\cdots\!00}a-\frac{57\!\cdots\!81}{33\!\cdots\!00}$, $\frac{18\!\cdots\!93}{94\!\cdots\!00}a^{14}+\frac{49\!\cdots\!71}{18\!\cdots\!00}a^{13}+\frac{16\!\cdots\!99}{37\!\cdots\!60}a^{12}-\frac{31\!\cdots\!41}{94\!\cdots\!00}a^{11}-\frac{33\!\cdots\!77}{94\!\cdots\!00}a^{10}+\frac{21\!\cdots\!47}{94\!\cdots\!00}a^{9}+\frac{21\!\cdots\!81}{94\!\cdots\!00}a^{8}+\frac{35\!\cdots\!09}{18\!\cdots\!00}a^{7}+\frac{34\!\cdots\!09}{94\!\cdots\!00}a^{6}-\frac{30\!\cdots\!87}{94\!\cdots\!00}a^{5}+\frac{11\!\cdots\!09}{94\!\cdots\!00}a^{4}-\frac{24\!\cdots\!79}{94\!\cdots\!00}a^{3}+\frac{98\!\cdots\!11}{94\!\cdots\!00}a^{2}-\frac{39\!\cdots\!97}{47\!\cdots\!00}a-\frac{47\!\cdots\!06}{59\!\cdots\!79}$, $\frac{19\!\cdots\!63}{20\!\cdots\!00}a^{14}+\frac{49\!\cdots\!87}{82\!\cdots\!00}a^{13}+\frac{85\!\cdots\!71}{20\!\cdots\!00}a^{12}+\frac{14\!\cdots\!43}{16\!\cdots\!00}a^{11}+\frac{33\!\cdots\!27}{82\!\cdots\!00}a^{10}+\frac{12\!\cdots\!27}{20\!\cdots\!00}a^{9}+\frac{62\!\cdots\!09}{11\!\cdots\!00}a^{8}-\frac{55\!\cdots\!61}{82\!\cdots\!00}a^{7}-\frac{25\!\cdots\!48}{73\!\cdots\!75}a^{6}-\frac{19\!\cdots\!39}{16\!\cdots\!00}a^{5}-\frac{12\!\cdots\!67}{82\!\cdots\!00}a^{4}-\frac{29\!\cdots\!31}{42\!\cdots\!00}a^{3}+\frac{21\!\cdots\!97}{82\!\cdots\!00}a^{2}-\frac{19\!\cdots\!57}{82\!\cdots\!00}a-\frac{87\!\cdots\!99}{41\!\cdots\!00}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 213844962215.18634 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 213844962215.18634 \cdot 15}{2\cdot\sqrt{704066412430281882891194976796875}}\cr\approx \mathstrut & 46.7350323606252 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 + 9*x^13 - 137*x^12 + 171*x^11 - 729*x^10 + 6821*x^9 - 18729*x^8 + 68841*x^7 + 73741*x^6 + 1162125*x^5 - 113439*x^4 + 6042802*x^3 - 2166687*x^2 + 1882230*x + 1784980);
 
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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2835.1, 5.1.17935225.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
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 ${\href{/padicField/2.2.0.1}{2} }^{7}{,}\,{\href{/padicField/2.1.0.1}{1} }$ R R R R $15$ $15$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $15$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(3\) Copy content Toggle raw display 3.15.20.65$x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$$3$$5$$20$$C_{15}$$[2]^{5}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
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}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
\(11\) Copy content Toggle raw display 11.15.12.5$x^{15} - 44 x^{10} + 484 x^{5} + 107811$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2835.3t2.a.a$2$ $ 3^{4} \cdot 5 \cdot 7 $ 3.1.2835.1 $S_3$ (as 3T2) $1$ $0$
* 2.4235.5t2.a.b$2$ $ 5 \cdot 7 \cdot 11^{2}$ 5.1.17935225.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.4235.5t2.a.a$2$ $ 5 \cdot 7 \cdot 11^{2}$ 5.1.17935225.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.343035.15t2.a.b$2$ $ 3^{4} \cdot 5 \cdot 7 \cdot 11^{2}$ 15.1.704066412430281882891194976796875.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.343035.15t2.a.d$2$ $ 3^{4} \cdot 5 \cdot 7 \cdot 11^{2}$ 15.1.704066412430281882891194976796875.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.343035.15t2.a.a$2$ $ 3^{4} \cdot 5 \cdot 7 \cdot 11^{2}$ 15.1.704066412430281882891194976796875.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.343035.15t2.a.c$2$ $ 3^{4} \cdot 5 \cdot 7 \cdot 11^{2}$ 15.1.704066412430281882891194976796875.1 $D_{15}$ (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 *.