Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-114223080x-283150929600\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-114223080xz^2-283150929600z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-148033111707x-13210245672082506\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-8520, 271800\right)\)
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| $\hat{h}(P)$ | ≈ | $4.0697304843631873930432752506$ |
Torsion generators
\( \left(-2640, 1320\right) \), \( \left(-5580, 427620\right) \)
Integral points
\( \left(-8520, 271800\right) \), \( \left(-8520, -263280\right) \), \( \left(-5580, 427620\right) \), \( \left(-5580, -422040\right) \), \( \left(-2640, 1320\right) \), \( \left(11760, -5880\right) \), \( \left(13230, 715470\right) \), \( \left(13230, -728700\right) \), \( \left(16962, 1622346\right) \), \( \left(16962, -1639308\right) \), \( \left(29100, 4571880\right) \), \( \left(29100, -4600980\right) \), \( \left(69560, 18085520\right) \), \( \left(69560, -18155080\right) \)
Invariants
| Conductor: | \( 82110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $60743509039087274201760000 $ | = | $2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 17^{8} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{164810665209657549410090824321}{60743509039087274201760000} \) | = | $2^{-8} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 13^{3} \cdot 17^{-8} \cdot 23^{-4} \cdot 37^{3} \cdot 1213^{3} \cdot 9397^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $3.6475266189092227497237897523\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $3.6475266189092227497237897523\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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||
| $abc$ quality: | $1.006028426914323\dots$ | |||
| Szpiro ratio: | $5.945183294623821\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $4.0697304843631873930432752506\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.047612309810607448036923694146\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Tamagawa product: | $ 4096 $ = $ 2^{3}\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{3}\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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| Torsion order: | $8$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 12.401233194699109664652645957 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 12.401233195 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.047612 \cdot 4.069730 \cdot 4096}{8^2} \approx 12.401233195$
Modular invariants
Modular form 82110.2.a.bs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29360128 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $23$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.96.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 656880 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 8 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 618249 & 8 \\ 618232 & 656873 \end{array}\right),\left(\begin{array}{rr} 574785 & 8 \\ 328 & 410725 \end{array}\right),\left(\begin{array}{rr} 574785 & 8 \\ 328 & 175 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 469209 & 4 \\ 375272 & 656841 \end{array}\right),\left(\begin{array}{rr} 656865 & 16 \\ 656864 & 17 \end{array}\right),\left(\begin{array}{rr} 199921 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 525513 & 16 \\ 525068 & 656105 \end{array}\right),\left(\begin{array}{rr} 9 & 16 \\ 218524 & 656105 \end{array}\right)$.
The torsion field $K:=\Q(E[656880])$ is a degree-$31107765182178263040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/656880\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \) |
| $5$ | split multiplicative | $6$ | \( 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 \) |
| $17$ | split multiplicative | $18$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 82110bt
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{15})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.44033523122176.42 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | split | nonsplit | ord | ord | split | ord | nonsplit | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 10 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.