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Properties

Label 15.1.467...723.1
Degree $15$
Signature $[1, 7]$
Discriminant $-4.676\times 10^{32}$
Root discriminant \(150.66\)
Ramified primes $3,7,31$
Class number $135$ (GRH)
Class group [3, 45] (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 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900)
 
gp: K = bnfinit(y^15 - 3*y^14 - 2*y^13 + 10*y^12 + 1234*y^11 - 2708*y^10 - 7364*y^9 + 21436*y^8 + 142132*y^7 - 818166*y^6 - 1239170*y^5 + 17611188*y^4 - 4641981*y^3 - 117043497*y^2 + 64960110*y + 322857900, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900)
 

\( x^{15} - 3 x^{14} - 2 x^{13} + 10 x^{12} + 1234 x^{11} - 2708 x^{10} - 7364 x^{9} + 21436 x^{8} + \cdots + 322857900 \) 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:   \(-467619805431628537196815659789723\) \(\medspace = -\,3^{7}\cdot 7^{10}\cdot 31^{14}\) 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:  \(150.66\)
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:  $3^{1/2}7^{2/3}31^{14/15}\approx 156.2776606679498$
Ramified primes:   \(3\), \(7\), \(31\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{18}a^{7}+\frac{1}{18}a^{6}-\frac{1}{18}a^{4}+\frac{5}{18}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{1116}a^{8}-\frac{7}{558}a^{7}+\frac{13}{186}a^{6}+\frac{17}{279}a^{5}-\frac{43}{279}a^{4}+\frac{67}{186}a^{3}+\frac{185}{372}a^{2}+\frac{13}{62}a+\frac{1}{31}$, $\frac{1}{1116}a^{9}+\frac{1}{186}a^{7}-\frac{1}{62}a^{6}+\frac{1}{31}a^{5}+\frac{17}{186}a^{4}+\frac{293}{1116}a^{3}-\frac{46}{93}a^{2}-\frac{34}{93}a+\frac{14}{31}$, $\frac{1}{3348}a^{10}-\frac{1}{3348}a^{9}-\frac{1}{3348}a^{8}+\frac{1}{279}a^{7}+\frac{95}{1674}a^{6}+\frac{37}{837}a^{5}-\frac{1}{108}a^{4}-\frac{23}{124}a^{3}-\frac{193}{1116}a^{2}+\frac{1}{186}a-\frac{7}{31}$, $\frac{1}{66960}a^{11}+\frac{1}{8370}a^{10}-\frac{1}{4185}a^{9}-\frac{7}{22320}a^{8}-\frac{631}{33480}a^{7}+\frac{407}{8370}a^{6}+\frac{895}{13392}a^{5}+\frac{931}{5580}a^{4}-\frac{47}{372}a^{3}-\frac{251}{1488}a^{2}+\frac{1589}{3720}a+\frac{5}{124}$, $\frac{1}{66960}a^{12}+\frac{1}{2480}a^{9}+\frac{13}{33480}a^{8}+\frac{77}{4185}a^{7}+\frac{1667}{66960}a^{6}+\frac{463}{16740}a^{5}+\frac{511}{16740}a^{4}-\frac{1865}{4464}a^{3}+\frac{347}{11160}a^{2}+\frac{319}{1860}a+\frac{13}{31}$, $\frac{1}{382274640}a^{13}-\frac{166}{23892165}a^{12}+\frac{551}{191137320}a^{11}-\frac{37397}{382274640}a^{10}-\frac{9479}{191137320}a^{9}-\frac{21433}{191137320}a^{8}+\frac{1624591}{382274640}a^{7}-\frac{62867}{8688060}a^{6}-\frac{39739}{6165720}a^{5}-\frac{532777}{2831664}a^{4}-\frac{886093}{63712440}a^{3}+\frac{1722779}{21237480}a^{2}-\frac{15679}{707916}a-\frac{5327}{117986}$, $\frac{1}{12\!\cdots\!80}a^{14}+\frac{101961826849093}{16\!\cdots\!60}a^{13}+\frac{48\!\cdots\!39}{80\!\cdots\!30}a^{12}+\frac{14\!\cdots\!59}{64\!\cdots\!40}a^{11}-\frac{18\!\cdots\!59}{64\!\cdots\!40}a^{10}-\frac{24\!\cdots\!83}{70\!\cdots\!20}a^{9}-\frac{13\!\cdots\!83}{29\!\cdots\!20}a^{8}-\frac{66\!\cdots\!47}{64\!\cdots\!40}a^{7}+\frac{14\!\cdots\!59}{32\!\cdots\!20}a^{6}-\frac{76\!\cdots\!81}{21\!\cdots\!80}a^{5}-\frac{55\!\cdots\!11}{43\!\cdots\!16}a^{4}+\frac{70\!\cdots\!59}{19\!\cdots\!60}a^{3}+\frac{51\!\cdots\!71}{14\!\cdots\!20}a^{2}-\frac{20\!\cdots\!81}{79\!\cdots\!40}a-\frac{16\!\cdots\!13}{34\!\cdots\!48}$ 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:  $2$, $3$

Class group and class number

$C_{3}\times C_{45}$, which has order $135$ (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:   $\frac{44\!\cdots\!49}{12\!\cdots\!80}a^{14}-\frac{22\!\cdots\!87}{10\!\cdots\!40}a^{13}+\frac{22\!\cdots\!57}{71\!\cdots\!60}a^{12}-\frac{21\!\cdots\!67}{32\!\cdots\!20}a^{11}+\frac{92\!\cdots\!53}{21\!\cdots\!80}a^{10}-\frac{21\!\cdots\!41}{93\!\cdots\!60}a^{9}+\frac{21\!\cdots\!71}{20\!\cdots\!40}a^{8}+\frac{17\!\cdots\!59}{21\!\cdots\!80}a^{7}+\frac{63\!\cdots\!79}{21\!\cdots\!80}a^{6}-\frac{13\!\cdots\!81}{32\!\cdots\!20}a^{5}+\frac{13\!\cdots\!17}{39\!\cdots\!56}a^{4}+\frac{13\!\cdots\!43}{21\!\cdots\!80}a^{3}-\frac{27\!\cdots\!11}{14\!\cdots\!20}a^{2}-\frac{39\!\cdots\!17}{23\!\cdots\!20}a+\frac{36\!\cdots\!49}{34\!\cdots\!48}$, $\frac{24\!\cdots\!69}{64\!\cdots\!40}a^{14}-\frac{81\!\cdots\!09}{21\!\cdots\!80}a^{13}+\frac{13\!\cdots\!11}{35\!\cdots\!80}a^{12}-\frac{81\!\cdots\!43}{32\!\cdots\!20}a^{11}+\frac{10\!\cdots\!31}{21\!\cdots\!80}a^{10}-\frac{51\!\cdots\!47}{11\!\cdots\!70}a^{9}+\frac{29\!\cdots\!97}{73\!\cdots\!30}a^{8}-\frac{21\!\cdots\!53}{12\!\cdots\!60}a^{7}+\frac{55\!\cdots\!59}{10\!\cdots\!40}a^{6}-\frac{23\!\cdots\!71}{32\!\cdots\!20}a^{5}+\frac{23\!\cdots\!69}{21\!\cdots\!80}a^{4}+\frac{41\!\cdots\!81}{53\!\cdots\!20}a^{3}-\frac{11\!\cdots\!53}{71\!\cdots\!60}a^{2}-\frac{37\!\cdots\!59}{11\!\cdots\!60}a+\frac{38\!\cdots\!43}{17\!\cdots\!24}$, $\frac{3787026211}{72\!\cdots\!20}a^{14}-\frac{29278103687}{72\!\cdots\!20}a^{13}+\frac{22900887109}{14\!\cdots\!40}a^{12}+\frac{13227175487}{48\!\cdots\!80}a^{11}+\frac{921046104809}{14\!\cdots\!24}a^{10}-\frac{63579801479129}{14\!\cdots\!40}a^{9}-\frac{1658135406821}{53\!\cdots\!20}a^{8}+\frac{12254686702261}{72\!\cdots\!12}a^{7}+\frac{109221963466427}{29\!\cdots\!48}a^{6}-\frac{97\!\cdots\!93}{14\!\cdots\!40}a^{5}+\frac{154876039771159}{22\!\cdots\!40}a^{4}+\frac{13\!\cdots\!99}{96\!\cdots\!16}a^{3}-\frac{17\!\cdots\!29}{53\!\cdots\!20}a^{2}-\frac{35\!\cdots\!93}{538868902871112}a+\frac{16\!\cdots\!21}{89811483811852}$, $\frac{17\!\cdots\!39}{80\!\cdots\!30}a^{14}-\frac{35\!\cdots\!41}{12\!\cdots\!48}a^{13}-\frac{13\!\cdots\!73}{64\!\cdots\!40}a^{12}-\frac{29\!\cdots\!61}{43\!\cdots\!16}a^{11}+\frac{48\!\cdots\!37}{12\!\cdots\!48}a^{10}+\frac{51\!\cdots\!49}{28\!\cdots\!80}a^{9}-\frac{44\!\cdots\!63}{15\!\cdots\!08}a^{8}+\frac{21\!\cdots\!11}{64\!\cdots\!40}a^{7}+\frac{47\!\cdots\!13}{64\!\cdots\!40}a^{6}-\frac{14\!\cdots\!61}{64\!\cdots\!40}a^{5}-\frac{19\!\cdots\!99}{12\!\cdots\!60}a^{4}+\frac{55\!\cdots\!61}{19\!\cdots\!80}a^{3}+\frac{26\!\cdots\!57}{23\!\cdots\!20}a^{2}-\frac{33\!\cdots\!71}{23\!\cdots\!12}a-\frac{20\!\cdots\!21}{55\!\cdots\!04}$, $\frac{19\!\cdots\!83}{64\!\cdots\!40}a^{14}-\frac{54\!\cdots\!37}{26\!\cdots\!30}a^{13}-\frac{16\!\cdots\!03}{12\!\cdots\!48}a^{12}-\frac{64\!\cdots\!95}{12\!\cdots\!48}a^{11}+\frac{11\!\cdots\!59}{32\!\cdots\!20}a^{10}+\frac{75\!\cdots\!77}{28\!\cdots\!80}a^{9}-\frac{33\!\cdots\!37}{12\!\cdots\!48}a^{8}-\frac{78\!\cdots\!01}{16\!\cdots\!60}a^{7}+\frac{20\!\cdots\!19}{64\!\cdots\!40}a^{6}-\frac{12\!\cdots\!69}{79\!\cdots\!40}a^{5}-\frac{10\!\cdots\!01}{10\!\cdots\!40}a^{4}+\frac{40\!\cdots\!19}{13\!\cdots\!52}a^{3}+\frac{10\!\cdots\!89}{11\!\cdots\!60}a^{2}-\frac{34\!\cdots\!33}{19\!\cdots\!60}a-\frac{15\!\cdots\!11}{43\!\cdots\!81}$, $\frac{30\!\cdots\!61}{28\!\cdots\!80}a^{14}+\frac{14\!\cdots\!53}{18\!\cdots\!92}a^{13}-\frac{20\!\cdots\!57}{10\!\cdots\!40}a^{12}-\frac{15\!\cdots\!33}{25\!\cdots\!80}a^{11}+\frac{13\!\cdots\!57}{93\!\cdots\!60}a^{10}+\frac{10\!\cdots\!93}{30\!\cdots\!60}a^{9}-\frac{67\!\cdots\!77}{28\!\cdots\!80}a^{8}-\frac{36\!\cdots\!51}{34\!\cdots\!80}a^{7}+\frac{75\!\cdots\!19}{31\!\cdots\!20}a^{6}+\frac{43\!\cdots\!47}{28\!\cdots\!80}a^{5}-\frac{41\!\cdots\!91}{31\!\cdots\!20}a^{4}-\frac{93\!\cdots\!27}{93\!\cdots\!60}a^{3}+\frac{15\!\cdots\!96}{64\!\cdots\!15}a^{2}+\frac{19\!\cdots\!17}{64\!\cdots\!15}a+\frac{15\!\cdots\!77}{86\!\cdots\!62}$, $\frac{24\!\cdots\!79}{25\!\cdots\!96}a^{14}-\frac{93\!\cdots\!19}{89\!\cdots\!70}a^{13}-\frac{51\!\cdots\!89}{14\!\cdots\!72}a^{12}-\frac{45\!\cdots\!91}{64\!\cdots\!40}a^{11}+\frac{70\!\cdots\!79}{71\!\cdots\!60}a^{10}-\frac{11\!\cdots\!83}{93\!\cdots\!60}a^{9}-\frac{90\!\cdots\!87}{16\!\cdots\!06}a^{8}-\frac{20\!\cdots\!03}{21\!\cdots\!80}a^{7}-\frac{92\!\cdots\!77}{79\!\cdots\!40}a^{6}-\frac{15\!\cdots\!57}{64\!\cdots\!40}a^{5}-\frac{71\!\cdots\!09}{21\!\cdots\!80}a^{4}+\frac{81\!\cdots\!73}{21\!\cdots\!80}a^{3}+\frac{45\!\cdots\!49}{14\!\cdots\!20}a^{2}-\frac{59\!\cdots\!11}{21\!\cdots\!20}a-\frac{13\!\cdots\!59}{34\!\cdots\!48}$ 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:  \( 45692282010.76435 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 45692282010.76435 \cdot 135}{2\cdot\sqrt{467619805431628537196815659789723}}\cr\approx \mathstrut & 110.278092936803 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900)
 
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 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900, 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 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900);
 
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 - 3*x^14 - 2*x^13 + 10*x^12 + 1234*x^11 - 2708*x^10 - 7364*x^9 + 21436*x^8 + 142132*x^7 - 818166*x^6 - 1239170*x^5 + 17611188*x^4 - 4641981*x^3 - 117043497*x^2 + 64960110*x + 322857900);
 
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

3.1.141267.1, 5.1.8311689.1

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 ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $15$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ R $15$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
\(7\) Copy content Toggle raw display 7.15.10.3$x^{15} + 98 x^{9} + 2401 x^{3} + 268912$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
\(31\) Copy content Toggle raw display 31.15.14.14$x^{15} + 372$$15$$1$$14$$C_{15}$$[\ ]_{15}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.141267.3t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 31^{2}$ 3.1.141267.1 $S_3$ (as 3T2) $1$ $0$
* 2.2883.5t2.a.a$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2883.5t2.a.b$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.141267.15t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 31^{2}$ 15.1.467619805431628537196815659789723.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.141267.15t2.a.c$2$ $ 3 \cdot 7^{2} \cdot 31^{2}$ 15.1.467619805431628537196815659789723.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.141267.15t2.a.d$2$ $ 3 \cdot 7^{2} \cdot 31^{2}$ 15.1.467619805431628537196815659789723.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.141267.15t2.a.b$2$ $ 3 \cdot 7^{2} \cdot 31^{2}$ 15.1.467619805431628537196815659789723.1 $D_{15}$ (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 *.