Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-70808x+2686731\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-70808xz^2+2686731z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1132923x+170817878\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(3, 1571\right)\) | \(\left(-49, 2481\right)\) |
$\hat{h}(P)$ | ≈ | $1.4234100016420918309372881377$ | $2.2655707192664055318393470394$ |
Torsion generators
\( \left(-283, 141\right) \), \( \left(245, -123\right) \)
Integral points
\( \left(-283, 141\right) \), \( \left(-205, 3027\right) \), \( \left(-205, -2823\right) \), \( \left(-49, 2481\right) \), \( \left(-49, -2433\right) \), \( \left(3, 1571\right) \), \( \left(3, -1575\right) \), \( \left(245, -123\right) \), \( \left(311, 3111\right) \), \( \left(311, -3423\right) \), \( \left(443, 7401\right) \), \( \left(443, -7845\right) \), \( \left(501, 9381\right) \), \( \left(501, -9883\right) \), \( \left(795, 20777\right) \), \( \left(795, -21573\right) \), \( \left(1445, 53277\right) \), \( \left(1445, -54723\right) \), \( \left(1895, 80727\right) \), \( \left(1895, -82623\right) \), \( \left(6053, 467421\right) \), \( \left(6053, -473475\right) \), \( \left(87563, 25866795\right) \), \( \left(87563, -25954359\right) \)
Invariants
Conductor: | \( 141570 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $19643227808490000 $ | = | $2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{6} \cdot 13^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{30400540561}{15210000} \) | = | $2^{-4} \cdot 3^{-2} \cdot 5^{-4} \cdot 13^{-2} \cdot 3121^{3}$ | 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: | $1.8191798717724702921537345427\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.070926091039230174425140135256\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9623782022558851\dots$ | |||
Szpiro ratio: | $3.8039360078928803\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.2027625957852113364366679737\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.34116219186502184471854010541\dots$ | 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: | $ 256 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2^{2}\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: | $4$ | 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^{(2)}(E,1)/2! $ ≈ $ 17.482584115222234779814964640 $ | 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
$\displaystyle 17.482584115 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.341162 \cdot 3.202763 \cdot 256}{4^2} \approx 17.482584115$
Modular invariants
Modular form 141570.2.a.de
For more coefficients, see the Downloads section to the right.
Modular degree: | 1310720 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$13$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2Cs | 4.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1033 & 15598 \\ 14058 & 4685 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9359 & 0 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 17156 & 17157 \end{array}\right),\left(\begin{array}{rr} 6007 & 4686 \\ 13794 & 12475 \end{array}\right),\left(\begin{array}{rr} 11705 & 5852 \\ 5852 & 8581 \end{array}\right),\left(\begin{array}{rr} 1563 & 1958 \\ 4708 & 10605 \end{array}\right),\left(\begin{array}{rr} 10297 & 6248 \\ 9988 & 7833 \end{array}\right),\left(\begin{array}{rr} 17153 & 8 \\ 17152 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$63764692992000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 141570bq
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 390b2, its twist by $33$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-33}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-33}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.8670998958336.5 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.3035957760000.3 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.8.86709989583360000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ss | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 4 | - | 2 | 2,2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-invariant(s) | 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.