Properties

Label 190575en1
Conductor 190575190575
Discriminant 1.766×1016-1.766\times 10^{16}
j-invariant 40967 \frac{4096}{7}
CM no
Rank 11
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+y=x3+45375x5199219y^2+y=x^3+45375x-5199219 Copy content Toggle raw display (homogenize, simplify)
y2z+yz2=x3+45375xz25199219z3y^2z+yz^2=x^3+45375xz^2-5199219z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3+726000x332750000y^2=x^3+726000x-332750000 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 0, 1, 45375, -5199219])
 
Copy content gp:E = ellinit([0, 0, 1, 45375, -5199219])
 
Copy content magma:E := EllipticCurve([0, 0, 1, 45375, -5199219]);
 
Copy content oscar:E = elliptic_curve([0, 0, 1, 45375, -5199219])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(1160888037577569625/3526953785738244,1411145224428645660864994361/209459302793936124267528)(1160888037577569625/3526953785738244, 1411145224428645660864994361/209459302793936124267528)39.53987328149786093516454688739.539873281497860935164546887\infty

Integral points

None

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: NN  =  190575 190575  = 325271123^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  17656788638671875-17656788638671875 = 136597116-1 \cdot 3^{6} \cdot 5^{9} \cdot 7 \cdot 11^{6}
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  40967 \frac{4096}{7}  = 212712^{12} \cdot 7^{-1}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.80244657458475660966071621551.8024465745847566096607162155
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.1528856404740587890184476919-1.1528856404740587890184476919
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.98029579262198060.9802957926219806
Szpiro ratio: σm\sigma_{m} ≈ 3.65618610201322273.6561861020132227

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 39.53987328149786093516454688739.539873281497860935164546887
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 0.204318646922375303243222800110.20431864692237530324322280011
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 2 2  = 1211 1\cdot2\cdot1\cdot1
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L'(E,1) ≈ 16.15746681671564481277861295316.157466816715644812778612953
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

16.157466817L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.20431939.53987321216.157466817\begin{aligned} 16.157466817 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.204319 \cdot 39.539873 \cdot 2}{1^2} \\ & \approx 16.157466817\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([0, 0, 1, 45375, -5199219]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([0, 0, 1, 45375, -5199219]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 190575.2.a.fc

q+2q2+2q4+q7q13+2q144q16+7q17+O(q20) q + 2 q^{2} + 2 q^{4} + q^{7} - q^{13} + 2 q^{14} - 4 q^{16} + 7 q^{17} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ actual modular form, use for small N [mf,F] = mffromell(E) Ser(mfcoefs(mf,20),q) \\ or just the series Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 1620000
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
33 11 I0I_0^{*} additive -1 2 6 0
55 22 IIIIII^{*} additive -1 2 9 0
77 11 I1I_{1} split multiplicative -1 1 1 1
1111 11 I0I_0^{*} additive -1 2 6 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
55 5B.4.2 5.12.0.2

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[2146, 495, 825, 1816], [1, 0, 10, 1], [769, 0, 0, 2309], [6, 13, 2255, 2191], [2099, 0, 0, 2309], [2301, 10, 2300, 11], [1748, 1815, 495, 824], [1, 10, 0, 1]] GL(2,Integers(2310)).subgroup(gens)
 
Copy content magma:Gens := [[2146, 495, 825, 1816], [1, 0, 10, 1], [769, 0, 0, 2309], [6, 13, 2255, 2191], [2099, 0, 0, 2309], [2301, 10, 2300, 11], [1748, 1815, 495, 824], [1, 10, 0, 1]]; sub<GL(2,Integers(2310))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 2310=235711 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 , index 4848, genus 11, and generators

(21464958251816),(10101),(769002309),(61322552191),(2099002309),(230110230011),(17481815495824),(11001)\left(\begin{array}{rr} 2146 & 495 \\ 825 & 1816 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 769 & 0 \\ 0 & 2309 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 2255 & 2191 \end{array}\right),\left(\begin{array}{rr} 2099 & 0 \\ 0 & 2309 \end{array}\right),\left(\begin{array}{rr} 2301 & 10 \\ 2300 & 11 \end{array}\right),\left(\begin{array}{rr} 1748 & 1815 \\ 495 & 824 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[2310])K:=\Q(E[2310]) is a degree-7664025600076640256000 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/2310Z)\GL_2(\Z/2310\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 good 22 38115=3257112 38115 = 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}
33 additive 66 21175=527112 21175 = 5^{2} \cdot 7 \cdot 11^{2}
55 additive 1414 7623=327112 7623 = 3^{2} \cdot 7 \cdot 11^{2}
77 split multiplicative 88 27225=3252112 27225 = 3^{2} \cdot 5^{2} \cdot 11^{2}
1111 additive 6262 1575=32527 1575 = 3^{2} \cdot 5^{2} \cdot 7

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5.
Its isogeny class 190575en consists of 2 curves linked by isogenies of degree 5.

Twists

The minimal quadratic twist of this elliptic curve is 175a1, its twist by 165165.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.140.1 Z/2Z\Z/2\Z not in database
44 4.0.136125.2 Z/5Z\Z/5\Z not in database
66 6.0.686000.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 deg 8 Z/3Z\Z/3\Z not in database
1010 10.2.17625605679827578125.1 Z/5Z\Z/5\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/10Z\Z/10\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.