Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+3695x+481775\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+3695xz^2+481775z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+4788693x+22405872294\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(43, 828\right)\) | \(\left(-7, 678\right)\) |
$\hat{h}(P)$ | ≈ | $0.37171114707447945699815574266$ | $0.98071481294231563023807732622$ |
Torsion generators
\( \left(-\frac{253}{4}, \frac{249}{8}\right) \)
Integral points
\( \left(-57, 328\right) \), \( \left(-57, -272\right) \), \( \left(-47, 478\right) \), \( \left(-47, -432\right) \), \( \left(-25, 624\right) \), \( \left(-25, -600\right) \), \( \left(-7, 678\right) \), \( \left(-7, -672\right) \), \( \left(43, 828\right) \), \( \left(43, -872\right) \), \( \left(69, 1000\right) \), \( \left(69, -1070\right) \), \( \left(83, 1128\right) \), \( \left(83, -1212\right) \), \( \left(119, 1560\right) \), \( \left(119, -1680\right) \), \( \left(145, 1950\right) \), \( \left(145, -2096\right) \), \( \left(213, 3208\right) \), \( \left(213, -3422\right) \), \( \left(343, 6328\right) \), \( \left(343, -6672\right) \), \( \left(443, 9228\right) \), \( \left(443, -9672\right) \), \( \left(893, 26328\right) \), \( \left(893, -27222\right) \), \( \left(2423, 118128\right) \), \( \left(2423, -120552\right) \), \( \left(5143, 366328\right) \), \( \left(5143, -371472\right) \), \( \left(5803, 439228\right) \), \( \left(5803, -445032\right) \), \( \left(24653, 3858618\right) \), \( \left(24653, -3883272\right) \)
Invariants
Conductor: | \( 86190 \) | = | $2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-102859146000000 $ | = | $-1 \cdot 2^{7} \cdot 3^{4} \cdot 5^{6} \cdot 13^{3} \cdot 17^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2539391358707}{46818000000} \) | = | $2^{-7} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{3} \cdot 17^{-2} \cdot 1949^{3}$ | 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: | $1.3691353466707354258964662942\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $0.72789800730535124188309443381\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9694317040910475\dots$ | |||
Szpiro ratio: | $3.4922676075508816\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.34953760432810557499196237121\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.44515776825415081195439200084\dots$ | 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: | $ 336 $ = $ 7\cdot2\cdot( 2 \cdot 3 )\cdot2\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)
|
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^{(2)}(E,1)/2! $ ≈ $ 13.070347908542558203950208302 $ | 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
$\displaystyle 13.070347909 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.445158 \cdot 0.349538 \cdot 336}{2^2} \approx 13.070347909$
Modular invariants
Modular form 86190.2.a.bx
For more coefficients, see the Downloads section to the right.
Modular degree: | 387072 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$13$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8840 = 2^{3} \cdot 5 \cdot 13 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3316 & 5529 \\ 1105 & 7736 \end{array}\right),\left(\begin{array}{rr} 8837 & 4 \\ 8836 & 5 \end{array}\right),\left(\begin{array}{rr} 684 & 1 \\ 2039 & 0 \end{array}\right),\left(\begin{array}{rr} 3537 & 4 \\ 7074 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 4419 & 0 \end{array}\right),\left(\begin{array}{rr} 3641 & 4 \\ 7282 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[8840])$ is a degree-$126138156318720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 86190cg
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.507946400.3 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | split | ord | ord | add | nonsplit | ord | ord | ord | ss | ord | ord | ord | ss |
$\lambda$-invariant(s) | 4 | 2 | 3 | 2 | 2 | - | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2,2 |
$\mu$-invariant(s) | 1 | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.