Properties

Label 17.1.348...321.1
Degree $17$
Signature $[1, 8]$
Discriminant $3.490\times 10^{25}$
Root discriminant \(31.81\)
Ramified prime $1559$
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 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337)
 
gp: K = bnfinit(y^17 - 2*y^16 - 3*y^14 + 24*y^13 + 18*y^12 - 138*y^10 - 203*y^9 - 75*y^8 + 352*y^7 + 588*y^6 + 384*y^5 + 132*y^4 + 493*y^3 + 1015*y^2 + 874*y + 337, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337)
 

\( x^{17} - 2 x^{16} - 3 x^{14} + 24 x^{13} + 18 x^{12} - 138 x^{10} - 203 x^{9} - 75 x^{8} + 352 x^{7} + 588 x^{6} + 384 x^{5} + 132 x^{4} + 493 x^{3} + 1015 x^{2} + 874 x + 337 \) 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:   \(34895459505131153638432321\) \(\medspace = 1559^{8}\) 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:  \(31.81\)
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:  $1559^{1/2}\approx 39.48417404479927$
Ramified primes:   \(1559\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{443}a^{15}-\frac{22}{443}a^{14}+\frac{5}{443}a^{13}+\frac{164}{443}a^{12}-\frac{115}{443}a^{11}+\frac{86}{443}a^{10}+\frac{18}{443}a^{9}+\frac{190}{443}a^{8}+\frac{128}{443}a^{7}+\frac{214}{443}a^{6}+\frac{197}{443}a^{5}+\frac{132}{443}a^{4}+\frac{206}{443}a^{3}+\frac{169}{443}a^{2}+\frac{90}{443}a+\frac{124}{443}$, $\frac{1}{62\!\cdots\!87}a^{16}-\frac{59\!\cdots\!16}{62\!\cdots\!87}a^{15}+\frac{25\!\cdots\!83}{62\!\cdots\!87}a^{14}-\frac{19\!\cdots\!64}{62\!\cdots\!87}a^{13}-\frac{16\!\cdots\!89}{62\!\cdots\!87}a^{12}-\frac{27\!\cdots\!86}{62\!\cdots\!87}a^{11}-\frac{47\!\cdots\!68}{62\!\cdots\!87}a^{10}+\frac{32\!\cdots\!68}{62\!\cdots\!87}a^{9}+\frac{18\!\cdots\!28}{62\!\cdots\!87}a^{8}+\frac{66\!\cdots\!42}{62\!\cdots\!87}a^{7}-\frac{16\!\cdots\!79}{62\!\cdots\!87}a^{6}-\frac{13\!\cdots\!77}{60\!\cdots\!29}a^{5}+\frac{14\!\cdots\!15}{62\!\cdots\!87}a^{4}-\frac{39\!\cdots\!71}{62\!\cdots\!87}a^{3}-\frac{21\!\cdots\!39}{62\!\cdots\!87}a^{2}+\frac{30\!\cdots\!28}{60\!\cdots\!29}a+\frac{32\!\cdots\!80}{62\!\cdots\!87}$ 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:   $\frac{80\!\cdots\!66}{62\!\cdots\!87}a^{16}-\frac{20\!\cdots\!22}{62\!\cdots\!87}a^{15}+\frac{59\!\cdots\!19}{62\!\cdots\!87}a^{14}-\frac{58\!\cdots\!47}{62\!\cdots\!87}a^{13}+\frac{16\!\cdots\!58}{62\!\cdots\!87}a^{12}+\frac{74\!\cdots\!47}{62\!\cdots\!87}a^{11}-\frac{16\!\cdots\!78}{62\!\cdots\!87}a^{10}-\frac{92\!\cdots\!04}{62\!\cdots\!87}a^{9}-\frac{10\!\cdots\!95}{62\!\cdots\!87}a^{8}+\frac{44\!\cdots\!52}{62\!\cdots\!87}a^{7}+\frac{25\!\cdots\!94}{62\!\cdots\!87}a^{6}+\frac{24\!\cdots\!23}{60\!\cdots\!29}a^{5}+\frac{49\!\cdots\!76}{62\!\cdots\!87}a^{4}+\frac{85\!\cdots\!48}{62\!\cdots\!87}a^{3}+\frac{46\!\cdots\!42}{62\!\cdots\!87}a^{2}+\frac{55\!\cdots\!67}{60\!\cdots\!29}a+\frac{24\!\cdots\!49}{62\!\cdots\!87}$, $\frac{10\!\cdots\!18}{62\!\cdots\!87}a^{16}-\frac{20\!\cdots\!62}{62\!\cdots\!87}a^{15}-\frac{43\!\cdots\!15}{62\!\cdots\!87}a^{14}-\frac{18\!\cdots\!89}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!28}{62\!\cdots\!87}a^{12}+\frac{14\!\cdots\!87}{62\!\cdots\!87}a^{11}-\frac{86\!\cdots\!05}{62\!\cdots\!87}a^{10}-\frac{14\!\cdots\!81}{62\!\cdots\!87}a^{9}-\frac{17\!\cdots\!85}{62\!\cdots\!87}a^{8}-\frac{20\!\cdots\!21}{62\!\cdots\!87}a^{7}+\frac{43\!\cdots\!30}{62\!\cdots\!87}a^{6}+\frac{47\!\cdots\!10}{60\!\cdots\!29}a^{5}+\frac{20\!\cdots\!94}{62\!\cdots\!87}a^{4}-\frac{23\!\cdots\!44}{62\!\cdots\!87}a^{3}+\frac{57\!\cdots\!31}{62\!\cdots\!87}a^{2}+\frac{86\!\cdots\!46}{60\!\cdots\!29}a+\frac{56\!\cdots\!30}{62\!\cdots\!87}$, $\frac{42\!\cdots\!26}{62\!\cdots\!87}a^{16}-\frac{13\!\cdots\!32}{62\!\cdots\!87}a^{15}+\frac{16\!\cdots\!90}{62\!\cdots\!87}a^{14}-\frac{23\!\cdots\!59}{62\!\cdots\!87}a^{13}+\frac{11\!\cdots\!90}{62\!\cdots\!87}a^{12}-\frac{63\!\cdots\!43}{62\!\cdots\!87}a^{11}+\frac{73\!\cdots\!21}{62\!\cdots\!87}a^{10}-\frac{40\!\cdots\!45}{62\!\cdots\!87}a^{9}-\frac{18\!\cdots\!07}{62\!\cdots\!87}a^{8}+\frac{22\!\cdots\!65}{62\!\cdots\!87}a^{7}+\frac{36\!\cdots\!18}{62\!\cdots\!87}a^{6}+\frac{16\!\cdots\!19}{60\!\cdots\!29}a^{5}-\frac{27\!\cdots\!79}{62\!\cdots\!87}a^{4}+\frac{24\!\cdots\!19}{62\!\cdots\!87}a^{3}+\frac{46\!\cdots\!35}{62\!\cdots\!87}a^{2}+\frac{41\!\cdots\!68}{60\!\cdots\!29}a+\frac{12\!\cdots\!90}{62\!\cdots\!87}$, $\frac{90\!\cdots\!41}{62\!\cdots\!87}a^{16}-\frac{28\!\cdots\!63}{62\!\cdots\!87}a^{15}+\frac{29\!\cdots\!46}{62\!\cdots\!87}a^{14}-\frac{46\!\cdots\!39}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!70}{62\!\cdots\!87}a^{12}-\frac{11\!\cdots\!21}{62\!\cdots\!87}a^{11}+\frac{13\!\cdots\!10}{62\!\cdots\!87}a^{10}-\frac{11\!\cdots\!12}{62\!\cdots\!87}a^{9}-\frac{43\!\cdots\!76}{62\!\cdots\!87}a^{8}+\frac{34\!\cdots\!27}{62\!\cdots\!87}a^{7}+\frac{28\!\cdots\!81}{62\!\cdots\!87}a^{6}+\frac{15\!\cdots\!98}{60\!\cdots\!29}a^{5}+\frac{39\!\cdots\!56}{62\!\cdots\!87}a^{4}+\frac{23\!\cdots\!55}{62\!\cdots\!87}a^{3}+\frac{43\!\cdots\!03}{62\!\cdots\!87}a^{2}+\frac{37\!\cdots\!08}{60\!\cdots\!29}a+\frac{17\!\cdots\!86}{62\!\cdots\!87}$, $\frac{13\!\cdots\!91}{62\!\cdots\!87}a^{16}-\frac{36\!\cdots\!18}{62\!\cdots\!87}a^{15}+\frac{85\!\cdots\!92}{62\!\cdots\!87}a^{14}+\frac{14\!\cdots\!89}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!07}{62\!\cdots\!87}a^{12}+\frac{72\!\cdots\!80}{62\!\cdots\!87}a^{11}-\frac{36\!\cdots\!72}{62\!\cdots\!87}a^{10}-\frac{14\!\cdots\!54}{62\!\cdots\!87}a^{9}-\frac{12\!\cdots\!60}{62\!\cdots\!87}a^{8}+\frac{12\!\cdots\!49}{62\!\cdots\!87}a^{7}+\frac{43\!\cdots\!07}{62\!\cdots\!87}a^{6}+\frac{24\!\cdots\!33}{60\!\cdots\!29}a^{5}-\frac{22\!\cdots\!34}{62\!\cdots\!87}a^{4}+\frac{18\!\cdots\!26}{62\!\cdots\!87}a^{3}+\frac{61\!\cdots\!92}{62\!\cdots\!87}a^{2}+\frac{54\!\cdots\!76}{60\!\cdots\!29}a+\frac{23\!\cdots\!11}{62\!\cdots\!87}$, $\frac{16\!\cdots\!96}{62\!\cdots\!87}a^{16}-\frac{15\!\cdots\!27}{62\!\cdots\!87}a^{15}+\frac{43\!\cdots\!76}{62\!\cdots\!87}a^{14}-\frac{62\!\cdots\!59}{62\!\cdots\!87}a^{13}+\frac{12\!\cdots\!36}{62\!\cdots\!87}a^{12}-\frac{32\!\cdots\!22}{62\!\cdots\!87}a^{11}+\frac{23\!\cdots\!94}{62\!\cdots\!87}a^{10}-\frac{24\!\cdots\!41}{62\!\cdots\!87}a^{9}+\frac{10\!\cdots\!35}{62\!\cdots\!87}a^{8}+\frac{30\!\cdots\!08}{62\!\cdots\!87}a^{7}-\frac{49\!\cdots\!31}{62\!\cdots\!87}a^{6}-\frac{15\!\cdots\!97}{60\!\cdots\!29}a^{5}-\frac{15\!\cdots\!45}{62\!\cdots\!87}a^{4}+\frac{11\!\cdots\!98}{62\!\cdots\!87}a^{3}-\frac{13\!\cdots\!21}{62\!\cdots\!87}a^{2}-\frac{25\!\cdots\!19}{60\!\cdots\!29}a-\frac{30\!\cdots\!93}{62\!\cdots\!87}$, $\frac{36\!\cdots\!57}{62\!\cdots\!87}a^{16}-\frac{16\!\cdots\!58}{62\!\cdots\!87}a^{15}+\frac{79\!\cdots\!06}{62\!\cdots\!87}a^{14}+\frac{60\!\cdots\!44}{62\!\cdots\!87}a^{13}-\frac{45\!\cdots\!78}{62\!\cdots\!87}a^{12}-\frac{46\!\cdots\!98}{62\!\cdots\!87}a^{11}-\frac{26\!\cdots\!59}{62\!\cdots\!87}a^{10}-\frac{74\!\cdots\!29}{62\!\cdots\!87}a^{9}+\frac{68\!\cdots\!69}{62\!\cdots\!87}a^{8}+\frac{13\!\cdots\!71}{62\!\cdots\!87}a^{7}+\frac{20\!\cdots\!50}{62\!\cdots\!87}a^{6}-\frac{36\!\cdots\!02}{60\!\cdots\!29}a^{5}-\frac{86\!\cdots\!72}{62\!\cdots\!87}a^{4}+\frac{34\!\cdots\!07}{62\!\cdots\!87}a^{3}+\frac{49\!\cdots\!60}{62\!\cdots\!87}a^{2}-\frac{48\!\cdots\!20}{60\!\cdots\!29}a-\frac{19\!\cdots\!30}{62\!\cdots\!87}$, $\frac{25\!\cdots\!56}{62\!\cdots\!87}a^{16}-\frac{59\!\cdots\!71}{62\!\cdots\!87}a^{15}+\frac{15\!\cdots\!78}{62\!\cdots\!87}a^{14}-\frac{60\!\cdots\!55}{62\!\cdots\!87}a^{13}+\frac{62\!\cdots\!08}{62\!\cdots\!87}a^{12}+\frac{22\!\cdots\!34}{62\!\cdots\!87}a^{11}+\frac{61\!\cdots\!00}{62\!\cdots\!87}a^{10}-\frac{37\!\cdots\!72}{62\!\cdots\!87}a^{9}-\frac{30\!\cdots\!70}{62\!\cdots\!87}a^{8}-\frac{12\!\cdots\!80}{62\!\cdots\!87}a^{7}+\frac{10\!\cdots\!75}{62\!\cdots\!87}a^{6}+\frac{89\!\cdots\!30}{60\!\cdots\!29}a^{5}+\frac{68\!\cdots\!91}{62\!\cdots\!87}a^{4}-\frac{22\!\cdots\!98}{62\!\cdots\!87}a^{3}+\frac{13\!\cdots\!35}{62\!\cdots\!87}a^{2}+\frac{17\!\cdots\!87}{60\!\cdots\!29}a+\frac{14\!\cdots\!20}{62\!\cdots\!87}$ 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:  \( 432952.23952 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 432952.23952 \cdot 1}{2\cdot\sqrt{34895459505131153638432321}}\cr\approx \mathstrut & 0.17803052664 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337)
 
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 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337, 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 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337);
 
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 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337);
 
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 $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $17$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $17$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(1559\) Copy content Toggle raw display $\Q_{1559}$$x$$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}$