Properties

Label 294.e1
Conductor 294294
Discriminant 1.614×1012-1.614\times 10^{12}
j-invariant 6329617441279936 -\frac{6329617441}{279936}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3+x26910x232261y^2+xy+y=x^3+x^2-6910x-232261 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3+x2z6910xz2232261z3y^2z+xyz+yz^2=x^3+x^2z-6910xz^2-232261z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x38955387x10702030122y^2=x^3-8955387x-10702030122 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 1, -6910, -232261])
 
Copy content gp:E = ellinit([1, 1, 1, -6910, -232261])
 
Copy content magma:E := EllipticCurve([1, 1, 1, -6910, -232261]);
 
Copy content oscar:E = elliptic_curve([1, 1, 1, -6910, -232261])
 
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

trivial

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

Invariants

Conductor: NN  =  294 294  = 23722 \cdot 3 \cdot 7^{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  =  1613775332736-1613775332736 = 1273778-1 \cdot 2^{7} \cdot 3^{7} \cdot 7^{8}
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  =  6329617441279936 -\frac{6329617441}{279936}  = 1273779673-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7 \cdot 967^{3}
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}} ≈ 1.10743163801431062341539007801.1074316380143106234153900780
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}} ≈ 0.18984179468923157998817841763-0.18984179468923157998817841763
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.0323352202486651.032335220248665
Szpiro ratio: σm\sigma_{m} ≈ 6.7227754432756396.722775443275639

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
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 = 0 0
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) = 11
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 ≈ 0.261142918288111058704410345890.26114291828811105870441034589
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 = 7 7  = 711 7\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) ≈ 1.82800042801677741093087242131.8280004280167774109308724213
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    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

1.828000428L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.2611431.0000007121.828000428\begin{aligned} 1.828000428 \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 0.261143 \cdot 1.000000 \cdot 7}{1^2} \\ & \approx 1.828000428\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([1, 1, 1, -6910, -232261]); 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([1, 1, 1, -6910, -232261]); 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   294.2.a.e

q+q2q3+q4+q5q6+q8+q9+q10+5q11q12q15+q164q17+q18+8q19+O(q20) q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 5 q^{11} - q^{12} - q^{15} + q^{16} - 4 q^{17} + q^{18} + 8 q^{19} + 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: 588
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: no
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 3 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))
22 77 I7I_{7} split multiplicative -1 1 7 7
33 11 I7I_{7} nonsplit multiplicative 1 1 7 7
77 11 IVIV^{*} additive 1 2 8 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
77 7B.1.6 7.48.0.4

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 = [[127, 14, 0, 1], [1, 0, 14, 1], [113, 14, 119, 99], [8, 5, 91, 57], [155, 14, 154, 15], [125, 70, 0, 53], [85, 14, 91, 99], [1, 14, 0, 1]] GL(2,Integers(168)).subgroup(gens)
 
Copy content magma:Gens := [[127, 14, 0, 1], [1, 0, 14, 1], [113, 14, 119, 99], [8, 5, 91, 57], [155, 14, 154, 15], [125, 70, 0, 53], [85, 14, 91, 99], [1, 14, 0, 1]]; sub<GL(2,Integers(168))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 168=2337 168 = 2^{3} \cdot 3 \cdot 7 , index 9696, genus 22, and generators

(1271401),(10141),(1131411999),(859157),(1551415415),(12570053),(85149199),(11401)\left(\begin{array}{rr} 127 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 113 & 14 \\ 119 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 155 & 14 \\ 154 & 15 \end{array}\right),\left(\begin{array}{rr} 125 & 70 \\ 0 & 53 \end{array}\right),\left(\begin{array}{rr} 85 & 14 \\ 91 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 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[168])K:=\Q(E[168]) is a degree-15482881548288 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/168Z)\GL_2(\Z/168\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 split multiplicative 44 147=372 147 = 3 \cdot 7^{2}
33 nonsplit multiplicative 44 98=272 98 = 2 \cdot 7^{2}
77 additive 2626 1 1

Isogenies

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

This curve has non-trivial cyclic isogenies of degree dd for d=d= 7.
Its isogeny class 294.e consists of 2 curves linked by isogenies of degree 7.

Twists

The minimal quadratic twist of this elliptic curve is 294.f1, its twist by 7-7.

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
22 Q(7)\Q(\sqrt{-7}) Z/7Z\Z/7\Z 2.0.7.1-1764.2-c1
33 3.1.1176.1 Z/2Z\Z/2\Z not in database
66 6.0.33191424.2 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
66 6.0.9680832.1 Z/14Z\Z/14\Z not in database
88 8.2.6805279152.1 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/2ZZ/14Z\Z/2\Z \oplus \Z/14\Z not in database
1616 deg 16 Z/21Z\Z/21\Z not in database
2121 21.3.378818692265664781682717625943.3 Z/7Z\Z/7\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

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split nonsplit ord add ord ss ord ord ord ord ord ord ss ord ord
λ\lambda-invariant(s) 1 0 4 - 0 0,0 0 0 0 0 0 0 0,0 0 0
μ\mu-invariant(s) 0 0 0 - 0 0,0 0 0 0 0 0 0 0,0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.