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Properties

Label 17.1.105...401.1
Degree $17$
Signature $[1, 8]$
Discriminant $1.058\times 10^{25}$
Root discriminant \(29.65\)
Ramified primes $17,79$
Class number $1$
Class group trivial
Galois group $D_{17}$ (as 17T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23)
 
gp: K = bnfinit(y^17 - 3*y^16 - 2*y^15 + 24*y^14 - 20*y^13 - 80*y^12 + 115*y^11 + 27*y^10 + 132*y^9 - 302*y^8 - 219*y^7 - 172*y^6 + 41*y^5 - 25*y^4 + 44*y^3 + 39*y^2 - 2*y - 23, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23)
 

\( x^{17} - 3 x^{16} - 2 x^{15} + 24 x^{14} - 20 x^{13} - 80 x^{12} + 115 x^{11} + 27 x^{10} + 132 x^{9} + \cdots - 23 \) Copy content Toggle raw display

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(10582983069512347410470401\) \(\medspace = 17^{8}\cdot 79^{8}\) 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:  \(29.65\)
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:  $17^{1/2}79^{1/2}\approx 36.64696440361739$
Ramified primes:   \(17\), \(79\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{85}a^{13}-\frac{1}{17}a^{12}+\frac{33}{85}a^{11}-\frac{2}{85}a^{10}-\frac{6}{85}a^{9}-\frac{2}{17}a^{8}-\frac{3}{17}a^{7}+\frac{21}{85}a^{6}-\frac{6}{85}a^{5}-\frac{28}{85}a^{4}+\frac{27}{85}a^{3}-\frac{6}{85}a^{2}+\frac{26}{85}a-\frac{6}{17}$, $\frac{1}{85}a^{14}+\frac{8}{85}a^{12}-\frac{7}{85}a^{11}-\frac{16}{85}a^{10}-\frac{8}{17}a^{9}+\frac{4}{17}a^{8}+\frac{31}{85}a^{7}+\frac{14}{85}a^{6}+\frac{27}{85}a^{5}-\frac{28}{85}a^{4}-\frac{41}{85}a^{3}-\frac{4}{85}a^{2}+\frac{3}{17}a+\frac{4}{17}$, $\frac{1}{85}a^{15}-\frac{1}{85}a^{12}-\frac{5}{17}a^{11}-\frac{41}{85}a^{10}-\frac{2}{5}a^{9}-\frac{5}{17}a^{8}-\frac{36}{85}a^{7}+\frac{29}{85}a^{6}-\frac{14}{85}a^{5}-\frac{38}{85}a^{4}-\frac{33}{85}a^{3}-\frac{1}{17}a^{2}+\frac{16}{85}a+\frac{36}{85}$, $\frac{1}{59\!\cdots\!65}a^{16}+\frac{19008147736664}{59\!\cdots\!65}a^{15}+\frac{1447083794532}{59\!\cdots\!65}a^{14}+\frac{719896244228}{11\!\cdots\!93}a^{13}-\frac{119627042156754}{11\!\cdots\!93}a^{12}-\frac{18\!\cdots\!17}{59\!\cdots\!65}a^{11}-\frac{766118369873373}{59\!\cdots\!65}a^{10}+\frac{24\!\cdots\!47}{59\!\cdots\!65}a^{9}-\frac{13\!\cdots\!24}{59\!\cdots\!65}a^{8}-\frac{131590314109049}{352171069540645}a^{7}-\frac{283626651046264}{59\!\cdots\!65}a^{6}-\frac{255764972524883}{59\!\cdots\!65}a^{5}+\frac{11\!\cdots\!18}{59\!\cdots\!65}a^{4}+\frac{10\!\cdots\!49}{59\!\cdots\!65}a^{3}-\frac{29\!\cdots\!62}{59\!\cdots\!65}a^{2}-\frac{28\!\cdots\!72}{59\!\cdots\!65}a+\frac{12100959331581}{352171069540645}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:  $8$
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:   $\frac{111478658530044}{59\!\cdots\!65}a^{16}-\frac{54221298425038}{11\!\cdots\!93}a^{15}-\frac{334022900148948}{59\!\cdots\!65}a^{14}+\frac{22\!\cdots\!71}{59\!\cdots\!65}a^{13}-\frac{757078264540436}{59\!\cdots\!65}a^{12}-\frac{82\!\cdots\!08}{59\!\cdots\!65}a^{11}+\frac{52\!\cdots\!76}{59\!\cdots\!65}a^{10}+\frac{47\!\cdots\!04}{59\!\cdots\!65}a^{9}+\frac{28\!\cdots\!83}{59\!\cdots\!65}a^{8}-\frac{28\!\cdots\!87}{59\!\cdots\!65}a^{7}-\frac{36\!\cdots\!28}{59\!\cdots\!65}a^{6}-\frac{11\!\cdots\!62}{11\!\cdots\!93}a^{5}-\frac{583707592600738}{352171069540645}a^{4}-\frac{12\!\cdots\!11}{59\!\cdots\!65}a^{3}+\frac{65\!\cdots\!87}{59\!\cdots\!65}a^{2}+\frac{21\!\cdots\!11}{59\!\cdots\!65}a+\frac{54\!\cdots\!28}{59\!\cdots\!65}$, $\frac{28400996240856}{59\!\cdots\!65}a^{16}-\frac{46534273060209}{59\!\cdots\!65}a^{15}-\frac{172526513650117}{59\!\cdots\!65}a^{14}+\frac{632334979306721}{59\!\cdots\!65}a^{13}+\frac{50670575297451}{11\!\cdots\!93}a^{12}-\frac{30\!\cdots\!66}{59\!\cdots\!65}a^{11}+\frac{913444639293284}{59\!\cdots\!65}a^{10}+\frac{807251447673325}{11\!\cdots\!93}a^{9}+\frac{30\!\cdots\!42}{59\!\cdots\!65}a^{8}+\frac{19\!\cdots\!61}{59\!\cdots\!65}a^{7}-\frac{21\!\cdots\!71}{59\!\cdots\!65}a^{6}-\frac{577355622674523}{352171069540645}a^{5}-\frac{11\!\cdots\!88}{59\!\cdots\!65}a^{4}-\frac{61\!\cdots\!08}{59\!\cdots\!65}a^{3}+\frac{283059615981251}{59\!\cdots\!65}a^{2}+\frac{23\!\cdots\!82}{59\!\cdots\!65}a+\frac{30\!\cdots\!63}{59\!\cdots\!65}$, $\frac{18378543522719}{11\!\cdots\!93}a^{16}-\frac{359897593361758}{59\!\cdots\!65}a^{15}+\frac{56770847623123}{59\!\cdots\!65}a^{14}+\frac{23\!\cdots\!62}{59\!\cdots\!65}a^{13}-\frac{38\!\cdots\!68}{59\!\cdots\!65}a^{12}-\frac{11\!\cdots\!72}{11\!\cdots\!93}a^{11}+\frac{17\!\cdots\!71}{59\!\cdots\!65}a^{10}-\frac{13\!\cdots\!86}{11\!\cdots\!93}a^{9}+\frac{21\!\cdots\!77}{11\!\cdots\!93}a^{8}-\frac{39\!\cdots\!24}{59\!\cdots\!65}a^{7}-\frac{20\!\cdots\!13}{59\!\cdots\!65}a^{6}+\frac{98\!\cdots\!31}{59\!\cdots\!65}a^{5}+\frac{16\!\cdots\!59}{59\!\cdots\!65}a^{4}+\frac{21\!\cdots\!20}{11\!\cdots\!93}a^{3}+\frac{89\!\cdots\!51}{59\!\cdots\!65}a^{2}+\frac{84\!\cdots\!74}{59\!\cdots\!65}a+\frac{36\!\cdots\!52}{59\!\cdots\!65}$, $\frac{100121164543256}{59\!\cdots\!65}a^{16}-\frac{64541455424329}{11\!\cdots\!93}a^{15}-\frac{186887191304074}{59\!\cdots\!65}a^{14}+\frac{26\!\cdots\!44}{59\!\cdots\!65}a^{13}-\frac{25\!\cdots\!71}{59\!\cdots\!65}a^{12}-\frac{88\!\cdots\!48}{59\!\cdots\!65}a^{11}+\frac{15\!\cdots\!78}{59\!\cdots\!65}a^{10}+\frac{185051197048199}{352171069540645}a^{9}+\frac{48\!\cdots\!13}{59\!\cdots\!65}a^{8}-\frac{59\!\cdots\!08}{11\!\cdots\!93}a^{7}-\frac{53\!\cdots\!61}{11\!\cdots\!93}a^{6}+\frac{24\!\cdots\!94}{11\!\cdots\!93}a^{5}+\frac{87\!\cdots\!78}{59\!\cdots\!65}a^{4}+\frac{66\!\cdots\!11}{59\!\cdots\!65}a^{3}-\frac{60\!\cdots\!66}{59\!\cdots\!65}a^{2}+\frac{651213886362992}{11\!\cdots\!93}a-\frac{21\!\cdots\!24}{59\!\cdots\!65}$, $\frac{21165752817846}{59\!\cdots\!65}a^{16}-\frac{210932235168217}{59\!\cdots\!65}a^{15}+\frac{352412094827461}{59\!\cdots\!65}a^{14}+\frac{10\!\cdots\!62}{59\!\cdots\!65}a^{13}-\frac{40\!\cdots\!76}{59\!\cdots\!65}a^{12}-\frac{248066633532684}{59\!\cdots\!65}a^{11}+\frac{17\!\cdots\!98}{59\!\cdots\!65}a^{10}-\frac{12\!\cdots\!03}{59\!\cdots\!65}a^{9}-\frac{30\!\cdots\!72}{11\!\cdots\!93}a^{8}-\frac{22\!\cdots\!08}{59\!\cdots\!65}a^{7}+\frac{89\!\cdots\!77}{11\!\cdots\!93}a^{6}+\frac{57\!\cdots\!18}{59\!\cdots\!65}a^{5}+\frac{11\!\cdots\!92}{59\!\cdots\!65}a^{4}-\frac{40\!\cdots\!51}{59\!\cdots\!65}a^{3}-\frac{20\!\cdots\!53}{59\!\cdots\!65}a^{2}+\frac{58\!\cdots\!03}{59\!\cdots\!65}a+\frac{58\!\cdots\!32}{59\!\cdots\!65}$, $\frac{142794476090386}{59\!\cdots\!65}a^{16}-\frac{366273918270612}{59\!\cdots\!65}a^{15}-\frac{432071012418687}{59\!\cdots\!65}a^{14}+\frac{32\!\cdots\!54}{59\!\cdots\!65}a^{13}-\frac{15\!\cdots\!97}{59\!\cdots\!65}a^{12}-\frac{11\!\cdots\!07}{59\!\cdots\!65}a^{11}+\frac{11\!\cdots\!56}{59\!\cdots\!65}a^{10}+\frac{74\!\cdots\!09}{59\!\cdots\!65}a^{9}+\frac{22\!\cdots\!02}{59\!\cdots\!65}a^{8}-\frac{31\!\cdots\!66}{59\!\cdots\!65}a^{7}-\frac{87\!\cdots\!62}{11\!\cdots\!93}a^{6}-\frac{26\!\cdots\!36}{352171069540645}a^{5}-\frac{18\!\cdots\!58}{59\!\cdots\!65}a^{4}-\frac{19\!\cdots\!33}{59\!\cdots\!65}a^{3}+\frac{20\!\cdots\!83}{59\!\cdots\!65}a^{2}+\frac{39\!\cdots\!07}{59\!\cdots\!65}a+\frac{606696253920667}{11\!\cdots\!93}$, $\frac{66133583863592}{59\!\cdots\!65}a^{16}-\frac{248887926911607}{59\!\cdots\!65}a^{15}-\frac{83304509567622}{59\!\cdots\!65}a^{14}+\frac{416988986011894}{11\!\cdots\!93}a^{13}-\frac{26\!\cdots\!72}{59\!\cdots\!65}a^{12}-\frac{68\!\cdots\!97}{59\!\cdots\!65}a^{11}+\frac{15\!\cdots\!33}{59\!\cdots\!65}a^{10}+\frac{22\!\cdots\!17}{59\!\cdots\!65}a^{9}-\frac{11\!\cdots\!99}{59\!\cdots\!65}a^{8}-\frac{18\!\cdots\!52}{59\!\cdots\!65}a^{7}-\frac{11\!\cdots\!22}{59\!\cdots\!65}a^{6}+\frac{41\!\cdots\!58}{59\!\cdots\!65}a^{5}+\frac{83\!\cdots\!68}{59\!\cdots\!65}a^{4}-\frac{21\!\cdots\!74}{59\!\cdots\!65}a^{3}-\frac{20\!\cdots\!23}{59\!\cdots\!65}a^{2}+\frac{18\!\cdots\!02}{11\!\cdots\!93}a-\frac{51025521005872}{11\!\cdots\!93}$, $\frac{139205541256083}{59\!\cdots\!65}a^{16}-\frac{25445026163686}{352171069540645}a^{15}-\frac{14941661667697}{352171069540645}a^{14}+\frac{34\!\cdots\!34}{59\!\cdots\!65}a^{13}-\frac{190872386081304}{352171069540645}a^{12}-\frac{11\!\cdots\!09}{59\!\cdots\!65}a^{11}+\frac{18\!\cdots\!81}{59\!\cdots\!65}a^{10}+\frac{15\!\cdots\!77}{59\!\cdots\!65}a^{9}+\frac{32\!\cdots\!10}{11\!\cdots\!93}a^{8}-\frac{77\!\cdots\!51}{11\!\cdots\!93}a^{7}-\frac{24\!\cdots\!17}{352171069540645}a^{6}+\frac{16\!\cdots\!56}{59\!\cdots\!65}a^{5}+\frac{19\!\cdots\!16}{59\!\cdots\!65}a^{4}-\frac{92\!\cdots\!02}{59\!\cdots\!65}a^{3}+\frac{15\!\cdots\!66}{59\!\cdots\!65}a^{2}+\frac{54\!\cdots\!26}{59\!\cdots\!65}a-\frac{13\!\cdots\!39}{11\!\cdots\!93}$ Copy content Toggle raw display
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:  \( 663161.006215 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 663161.006215 \cdot 1}{2\cdot\sqrt{10582983069512347410470401}}\cr\approx \mathstrut & 0.495169490405 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23)
 
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^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23, 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^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23);
 
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^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23);
 
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

$D_{17}$ (as 17T2):

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 34
The 10 conjugacy class representatives for $D_{17}$
Character table for $D_{17}$

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 $17$ $17$ ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $17$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $17$ R $17$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $17$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $17$ $17$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
\(79\) Copy content Toggle raw display $\Q_{79}$$x + 76$$1$$1$$0$Trivial$[\ ]$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} + 237$$2$$1$$1$$C_2$$[\ ]_{2}$