Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-1854732x+974589968\) | (homogenize, simplify) |
\(y^2z=x^3-1854732xz^2+974589968z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1854732x+974589968\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 487872 \) | = | $2^{6} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1983209476532207616 $ | = | $-1 \cdot 2^{18} \cdot 3^{12} \cdot 7^{6} \cdot 11^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{30515071121161}{85766121} \) | = | $-1 \cdot 3^{-6} \cdot 7^{-6} \cdot 11 \cdot 14051^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.3832379458389388140382274310\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.39456181853190424687109936736\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9949069283037906\dots$ | |||
Szpiro ratio: | $4.19293051458632\dots$ |
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: | $0.26317804601189079975767247247\dots$ | 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: | $ 8 $ = $ 2\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)
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Torsion order: | $1$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.1054243680951263980613797797 $ | 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
$\displaystyle 2.105424368 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.263178 \cdot 1.000000 \cdot 8}{1^2} \approx 2.105424368$
Modular invariants
Modular form 487872.2.a.jy
For more coefficients, see the Downloads section to the right.
Modular degree: | 10616832 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{8}^{*}$ | Additive | -1 | 6 | 18 | 0 |
$3$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$7$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$11$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 308 = 2^{2} \cdot 7 \cdot 11 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 307 & 304 \\ 306 & 299 \end{array}\right),\left(\begin{array}{rr} 45 & 4 \\ 90 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 303 & 301 \end{array}\right),\left(\begin{array}{rr} 307 & 227 \\ 0 & 153 \end{array}\right),\left(\begin{array}{rr} 57 & 4 \\ 114 & 9 \end{array}\right),\left(\begin{array}{rr} 305 & 4 \\ 304 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[308])$ is a degree-$638668800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/308\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 487872jy consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2541g1, its twist by $24$.
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 $0$.