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Properties

Label 19.1.101...431.1
Degree $19$
Signature $[1, 9]$
Discriminant $-1.016\times 10^{29}$
Root discriminant \(33.63\)
Ramified primes $3,557$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053)
 
gp: K = bnfinit(y^19 - 5*y^18 + 11*y^17 - 10*y^16 + 3*y^15 + 3*y^14 - 80*y^13 + 481*y^12 - 1180*y^11 + 1220*y^10 + 888*y^9 - 5070*y^8 + 8854*y^7 - 9440*y^6 + 6893*y^5 - 3748*y^4 + 2592*y^3 - 2790*y^2 + 2673*y - 1053, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053)
 

\( x^{19} - 5 x^{18} + 11 x^{17} - 10 x^{16} + 3 x^{15} + 3 x^{14} - 80 x^{13} + 481 x^{12} - 1180 x^{11} + \cdots - 1053 \) Copy content Toggle raw display

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-101575284882268140616515967431\) \(\medspace = -\,3^{9}\cdot 557^{9}\) 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:  \(33.63\)
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:  $3^{1/2}557^{1/2}\approx 40.87786687193939$
Ramified primes:   \(3\), \(557\) 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:  \(\Q(\sqrt{-1671}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}-\frac{1}{3}a$, $\frac{1}{9}a^{8}+\frac{1}{3}a^{4}-\frac{4}{9}a^{2}$, $\frac{1}{9}a^{9}+\frac{2}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{2}{9}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{11}-\frac{1}{27}a^{9}+\frac{1}{9}a^{7}+\frac{2}{27}a^{5}+\frac{1}{3}a^{4}+\frac{4}{27}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{10}+\frac{2}{27}a^{6}-\frac{5}{27}a^{4}+\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{81}a^{13}+\frac{1}{81}a^{11}+\frac{1}{27}a^{10}-\frac{2}{81}a^{9}-\frac{1}{27}a^{8}+\frac{8}{81}a^{7}-\frac{1}{9}a^{6}-\frac{1}{81}a^{5}-\frac{4}{27}a^{4}+\frac{11}{81}a^{3}+\frac{7}{27}a^{2}+\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{81}a^{14}+\frac{1}{81}a^{12}-\frac{2}{81}a^{10}-\frac{1}{81}a^{8}+\frac{1}{9}a^{7}-\frac{1}{81}a^{6}+\frac{1}{9}a^{5}+\frac{38}{81}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}$, $\frac{1}{243}a^{15}+\frac{1}{243}a^{14}+\frac{1}{243}a^{13}+\frac{1}{243}a^{12}-\frac{2}{243}a^{11}-\frac{2}{243}a^{10}-\frac{10}{243}a^{9}+\frac{8}{243}a^{8}+\frac{35}{243}a^{7}+\frac{35}{243}a^{6}-\frac{34}{243}a^{5}-\frac{61}{243}a^{4}-\frac{8}{27}a^{3}+\frac{11}{27}a^{2}-\frac{1}{3}$, $\frac{1}{3159}a^{16}-\frac{2}{3159}a^{15}+\frac{16}{3159}a^{14}+\frac{10}{3159}a^{13}+\frac{1}{243}a^{12}+\frac{43}{3159}a^{11}+\frac{158}{3159}a^{10}-\frac{67}{3159}a^{9}-\frac{70}{3159}a^{8}-\frac{298}{3159}a^{7}+\frac{302}{3159}a^{6}+\frac{245}{3159}a^{5}+\frac{406}{1053}a^{4}+\frac{203}{1053}a^{3}+\frac{25}{351}a^{2}+\frac{10}{117}a-\frac{1}{3}$, $\frac{1}{161109}a^{17}+\frac{16}{161109}a^{16}-\frac{37}{53703}a^{15}-\frac{61}{53703}a^{14}-\frac{161}{53703}a^{13}-\frac{653}{53703}a^{12}-\frac{290}{161109}a^{11}-\frac{122}{161109}a^{10}+\frac{424}{53703}a^{9}+\frac{2878}{53703}a^{8}-\frac{331}{53703}a^{7}+\frac{34}{243}a^{6}-\frac{9530}{161109}a^{5}+\frac{70828}{161109}a^{4}+\frac{4753}{17901}a^{3}+\frac{3028}{17901}a^{2}-\frac{11}{117}a+\frac{52}{153}$, $\frac{1}{1302244047}a^{18}+\frac{428}{144693783}a^{17}-\frac{80209}{1302244047}a^{16}+\frac{793007}{434081349}a^{15}+\frac{1936471}{434081349}a^{14}-\frac{63280}{33390873}a^{13}+\frac{313963}{100172619}a^{12}-\frac{7269046}{434081349}a^{11}-\frac{67681507}{1302244047}a^{10}-\frac{23466806}{434081349}a^{9}+\frac{18566036}{434081349}a^{8}+\frac{1014310}{33390873}a^{7}-\frac{24995450}{1302244047}a^{6}-\frac{59695478}{434081349}a^{5}+\frac{630458807}{1302244047}a^{4}+\frac{2484602}{5359029}a^{3}-\frac{51382954}{144693783}a^{2}-\frac{6634339}{16077087}a+\frac{520313}{1236699}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:   $\frac{2245027}{1302244047}a^{18}-\frac{1377262}{144693783}a^{17}+\frac{22557497}{1302244047}a^{16}-\frac{708839}{144693783}a^{15}-\frac{7222558}{434081349}a^{14}+\frac{276232}{48231261}a^{13}-\frac{199041563}{1302244047}a^{12}+\frac{132135868}{144693783}a^{11}-\frac{2557216489}{1302244047}a^{10}+\frac{126933494}{144693783}a^{9}+\frac{1748164948}{434081349}a^{8}-\frac{497693552}{48231261}a^{7}+\frac{1263225316}{100172619}a^{6}-\frac{382805542}{48231261}a^{5}-\frac{2114394868}{1302244047}a^{4}+\frac{3742811}{412233}a^{3}-\frac{1365238492}{144693783}a^{2}+\frac{94370168}{16077087}a-\frac{4057633}{1236699}$, $\frac{406838}{144693783}a^{18}-\frac{16183489}{1302244047}a^{17}+\frac{25830203}{1302244047}a^{16}-\frac{1763357}{434081349}a^{15}-\frac{3456923}{434081349}a^{14}-\frac{694141}{434081349}a^{13}-\frac{99833692}{434081349}a^{12}+\frac{1593348335}{1302244047}a^{11}-\frac{2996263279}{1302244047}a^{10}+\frac{342032915}{434081349}a^{9}+\frac{2037355205}{434081349}a^{8}-\frac{4824433418}{434081349}a^{7}+\frac{5858425957}{434081349}a^{6}-\frac{13216639594}{1302244047}a^{5}+\frac{6674182718}{1302244047}a^{4}-\frac{37093100}{16077087}a^{3}+\frac{437696276}{144693783}a^{2}-\frac{7291514}{1786343}a+\frac{1736531}{1236699}$, $\frac{253148}{76602591}a^{18}-\frac{2142689}{144693783}a^{17}+\frac{34669133}{1302244047}a^{16}-\frac{491263}{33390873}a^{15}-\frac{1517222}{434081349}a^{14}+\frac{131843}{33390873}a^{13}-\frac{349610864}{1302244047}a^{12}+\frac{622474220}{434081349}a^{11}-\frac{3929996398}{1302244047}a^{10}+\frac{826236733}{434081349}a^{9}+\frac{1951885688}{434081349}a^{8}-\frac{6175771847}{434081349}a^{7}+\frac{1497929795}{76602591}a^{6}-\frac{7545034148}{434081349}a^{5}+\frac{13080722648}{1302244047}a^{4}-\frac{770296021}{144693783}a^{3}+\frac{742806848}{144693783}a^{2}-\frac{1849121}{315237}a+\frac{5710457}{1236699}$, $\frac{194}{161109}a^{18}-\frac{776}{161109}a^{17}+\frac{1556}{161109}a^{16}-\frac{415}{53703}a^{15}+\frac{74}{17901}a^{14}+\frac{2}{243}a^{13}-\frac{15529}{161109}a^{12}+\frac{4613}{9477}a^{11}-\frac{167017}{161109}a^{10}+\frac{52042}{53703}a^{9}+\frac{2131}{1989}a^{8}-\frac{262618}{53703}a^{7}+\frac{1345769}{161109}a^{6}-\frac{1373501}{161109}a^{5}+\frac{909239}{161109}a^{4}-\frac{14555}{5967}a^{3}+\frac{34247}{17901}a^{2}-\frac{5060}{1989}a+\frac{353}{153}$, $\frac{290465}{434081349}a^{18}+\frac{23606}{144693783}a^{17}-\frac{3994465}{434081349}a^{16}+\frac{303121}{11130291}a^{15}-\frac{3611783}{144693783}a^{14}+\frac{1356740}{144693783}a^{13}-\frac{18545452}{434081349}a^{12}+\frac{347050}{11130291}a^{11}+\frac{361026104}{434081349}a^{10}-\frac{138542203}{48231261}a^{9}+\frac{571401761}{144693783}a^{8}+\frac{28897390}{144693783}a^{7}-\frac{4542944455}{434081349}a^{6}+\frac{1033309832}{48231261}a^{5}-\frac{10595745694}{434081349}a^{4}+\frac{2520860084}{144693783}a^{3}-\frac{28150219}{5359029}a^{2}-\frac{2531954}{1236699}a+\frac{1089784}{412233}$, $\frac{194}{161109}a^{18}-\frac{776}{161109}a^{17}+\frac{1556}{161109}a^{16}-\frac{415}{53703}a^{15}+\frac{74}{17901}a^{14}+\frac{2}{243}a^{13}-\frac{15529}{161109}a^{12}+\frac{4613}{9477}a^{11}-\frac{167017}{161109}a^{10}+\frac{52042}{53703}a^{9}+\frac{2131}{1989}a^{8}-\frac{262618}{53703}a^{7}+\frac{1345769}{161109}a^{6}-\frac{1373501}{161109}a^{5}+\frac{909239}{161109}a^{4}-\frac{8588}{5967}a^{3}-\frac{19456}{17901}a^{2}+\frac{2896}{1989}a+\frac{47}{153}$, $\frac{156175}{100172619}a^{18}-\frac{12834241}{1302244047}a^{17}+\frac{32766751}{1302244047}a^{16}-\frac{10663559}{434081349}a^{15}-\frac{28456}{11130291}a^{14}+\frac{4847834}{434081349}a^{13}-\frac{145765961}{1302244047}a^{12}+\frac{1199630408}{1302244047}a^{11}-\frac{266051672}{100172619}a^{10}+\frac{1317866165}{434081349}a^{9}+\frac{282965042}{144693783}a^{8}-\frac{5058850577}{434081349}a^{7}+\frac{25074823822}{1302244047}a^{6}-\frac{24764010793}{1302244047}a^{5}+\frac{16439622340}{1302244047}a^{4}-\frac{898602001}{144693783}a^{3}+\frac{747627193}{144693783}a^{2}-\frac{12141715}{1786343}a+\frac{6477520}{1236699}$, $\frac{76538}{100172619}a^{18}-\frac{609731}{144693783}a^{17}+\frac{786109}{100172619}a^{16}-\frac{1780010}{434081349}a^{15}-\frac{573553}{434081349}a^{14}-\frac{1039988}{434081349}a^{13}-\frac{83759752}{1302244047}a^{12}+\frac{169768072}{434081349}a^{11}-\frac{1151742266}{1302244047}a^{10}+\frac{15357535}{25534197}a^{9}+\frac{32303126}{25534197}a^{8}-\frac{1666019131}{434081349}a^{7}+\frac{7063602041}{1302244047}a^{6}-\frac{2124212620}{434081349}a^{5}+\frac{4426537708}{1302244047}a^{4}-\frac{27846064}{11130291}a^{3}+\frac{346637422}{144693783}a^{2}-\frac{41844076}{16077087}a+\frac{1474738}{1236699}$, $\frac{30167360}{1302244047}a^{18}-\frac{31363669}{434081349}a^{17}+\frac{24064966}{1302244047}a^{16}+\frac{124084066}{434081349}a^{15}-\frac{122024392}{434081349}a^{14}-\frac{96734486}{434081349}a^{13}-\frac{1859474611}{1302244047}a^{12}+\frac{1161346543}{144693783}a^{11}-\frac{6965635406}{1302244047}a^{10}-\frac{12267839923}{434081349}a^{9}+\frac{29746652116}{434081349}a^{8}-\frac{14844362749}{434081349}a^{7}-\frac{107417673898}{1302244047}a^{6}+\frac{64550291047}{434081349}a^{5}-\frac{94024211630}{1302244047}a^{4}-\frac{214282169}{11130291}a^{3}+\frac{24170956}{144693783}a^{2}+\frac{314037989}{5359029}a-\frac{41025089}{1236699}$ 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:  \( 191261859.157 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 191261859.157 \cdot 1}{2\cdot\sqrt{101575284882268140616515967431}}\cr\approx \mathstrut & 9.15910765549 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053)
 
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^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053, 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^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053);
 
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^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053);
 
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

$D_{19}$ (as 19T2):

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 38
The 11 conjugacy class representatives for $D_{19}$
Character table for $D_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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: data not computed
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 $19$ R $19$ $19$ $19$ ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{9}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $19$ $19$ ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $19$ $19$ $19$ $19$ ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$Trivial$[\ ]$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
\(557\) Copy content Toggle raw display $\Q_{557}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$