Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-302204x+4052833048\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-302204xz^2+4052833048z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4835259x+259376479830\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{6629}{4}, \frac{6625}{8}\right) \)
Integral points
None
Invariants
| Conductor: | \( 439569 \) | = | $3^{2} \cdot 13^{2} \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-7093765959418973848689 $ | = | $-1 \cdot 3^{6} \cdot 13^{6} \cdot 17^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{35937}{83521} \) | = | $-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}$ | 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: | $2.8717513741515877728917006439\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.37663612094134348095743300428\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.1807067659885138\dots$ | |||
| Szpiro ratio: | $4.445934507869084\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.10663567119851695099381984046\dots$ | 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: | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ | 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: | $2$ | 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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 3.4123414783525424318022348946 $ | 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
$\displaystyle 3.412341478 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.106636 \cdot 1.000000 \cdot 32}{2^2} \approx 3.412341478$
Modular invariants
Modular form 439569.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21233664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $13$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $17$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 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 |
|---|---|---|
| $2$ | 2B | 16.48.0.76 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 42432 = 2^{6} \cdot 3 \cdot 13 \cdot 17 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 15406 & 12831 \\ 21411 & 18994 \end{array}\right),\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14143 & 0 \\ 0 & 42431 \end{array}\right),\left(\begin{array}{rr} 30655 & 24414 \\ 32526 & 12247 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 124 \\ 18164 & 5097 \end{array}\right),\left(\begin{array}{rr} 39167 & 0 \\ 0 & 42431 \end{array}\right),\left(\begin{array}{rr} 29951 & 39156 \\ 0 & 42431 \end{array}\right),\left(\begin{array}{rr} 57 & 16 \\ 36080 & 40649 \end{array}\right),\left(\begin{array}{rr} 42369 & 64 \\ 42368 & 65 \end{array}\right)$.
The torsion field $K:=\Q(E[42432])$ is a degree-$403642100219904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42432\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 439569be
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17a1, its twist by $-663$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.