Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-542310927885x-153716421029442075\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-542310927885xz^2-153716421029442075z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-8676974846163x-9837859622859138962\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{33207572758423854811975269875379}{27017205823583125595369929}, \frac{142892120852881449824659217631655117935098287728}{140430242898527712539642085615945907717}\right)\) |
$\hat{h}(P)$ | ≈ | $69.213841510191285208381785485$ |
Torsion generators
\( \left(-\frac{1700685}{4}, \frac{1700685}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 471510 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $57431176295161500000000000 $ | = | $2^{11} \cdot 3^{10} \cdot 5^{12} \cdot 13^{7} \cdot 31 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{5012808770744123733046717639129}{16321500000000000} \) | = | $2^{-11} \cdot 3^{-4} \cdot 5^{-12} \cdot 13^{-1} \cdot 31^{-1} \cdot 1249^{3} \cdot 13702441^{3}$ | 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: | $5.0499849033907375051994468822\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $3.2182040803259142914750805430\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $69.213841510191285208381785485\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.0055633638695657054556359955784\dots$ | 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: | $ 8 $ = $ 1\cdot2\cdot2\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])
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Analytic order of Ш: | $4$ = $2^2$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.0804942810531619132386056530 $ | 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
$\displaystyle 3.080494281 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.005563 \cdot 69.213842 \cdot 8}{2^2} \approx 3.080494281$
Modular invariants
Modular form 471510.2.a.m
For more coefficients, see the Downloads section to the right.
Modular degree: | 2543321088 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{11}$ | Non-split multiplicative | 1 | 1 | 11 | 11 |
$3$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$13$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$31$ | $1$ | $I_{1}$ | 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 \( 48360 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 48353 & 8 \\ 48352 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1232 & 48357 \\ 47115 & 16118 \end{array}\right),\left(\begin{array}{rr} 18136 & 10083 \\ 2019 & 34288 \end{array}\right),\left(\begin{array}{rr} 32239 & 0 \\ 0 & 48359 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5728 & 3 \\ 17685 & 32242 \end{array}\right),\left(\begin{array}{rr} 22168 & 46353 \\ 26187 & 10054 \end{array}\right),\left(\begin{array}{rr} 29017 & 16128 \\ 3228 & 16153 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 48354 & 48355 \end{array}\right)$.
The torsion field $K:=\Q(E[48360])$ is a degree-$17251247849472000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 471510m
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 12090h4, its twist by $-39$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.