Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-47858x-3570703\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-47858xz^2-3570703z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-765723x-229290698\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-123, 727\right)\) |
$\hat{h}(P)$ | ≈ | $2.8234663229278357559045634026$ |
Torsion generators
\( \left(-159, 79\right) \)
Integral points
\( \left(-159, 79\right) \), \( \left(-123, 727\right) \), \( \left(-123, -605\right) \), \( \left(625, 14191\right) \), \( \left(625, -14817\right) \)
Invariants
Conductor: | \( 136710 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $1470182262312960 $ | = | $2^{12} \cdot 3^{9} \cdot 5 \cdot 7^{6} \cdot 31 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{141339344329}{17141760} \) | = | $2^{-12} \cdot 3^{-3} \cdot 5^{-1} \cdot 31^{-1} \cdot 5209^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.6408413107847469056267835974\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.11858009192303540737648460722\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.8234663229278357559045634026\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.32534719144095093837147743436\dots$ | 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);
|
Tamagawa product: | $ 48 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot1\cdot2\cdot1 $ | 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)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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) $ ≈ $ 11.023282059512164601378465708 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 11.023282060 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.325347 \cdot 2.823466 \cdot 48}{2^2} \approx 11.023282060$
Modular invariants
Modular form 136710.2.a.eu
For more coefficients, see the Downloads section to the right.
Modular degree: | 884736 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $2$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$31$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 26040 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 26033 & 8 \\ 26032 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 11908 & 7441 \\ 20111 & 11166 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 8672 & 3717 \\ 23555 & 18598 \end{array}\right),\left(\begin{array}{rr} 22793 & 19068 \\ 7910 & 17207 \end{array}\right),\left(\begin{array}{rr} 7204 & 7441 \\ 19103 & 11166 \end{array}\right),\left(\begin{array}{rr} 16283 & 12558 \\ 20930 & 4187 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 26034 & 26035 \end{array}\right),\left(\begin{array}{rr} 14879 & 0 \\ 0 & 26039 \end{array}\right)$.
The torsion field $K:=\Q(E[26040])$ is a degree-$1327019065344000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/26040\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 136710.eu
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 930.b3, its twist by $21$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{465}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-3255}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{465})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | add | nonsplit | add | ord | ord | ord | ord | ord | ord | split | ord | ord | ord | ss |
$\lambda$-invariant(s) | 13 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.