Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-4285980x+8468410275\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-4285980xz^2+8468410275z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-5554630107x+395118813680694\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-270, 98145\right)\) |
$\hat{h}(P)$ | ≈ | $1.3279495871643759595575203249$ |
Torsion generators
\( \left(-\frac{10881}{4}, \frac{10881}{8}\right) \)
Integral points
\( \left(-270, 98145\right) \), \( \left(-270, -97875\right) \), \( \left(4590, 290115\right) \), \( \left(4590, -294705\right) \)
Invariants
Conductor: | \( 416955 \) | = | $3 \cdot 5 \cdot 7 \cdot 11 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-25944211427844840235875 $ | = | $-1 \cdot 3^{16} \cdot 5^{3} \cdot 7 \cdot 11^{4} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{185077034913624841}{551466161890875} \) | = | $-1 \cdot 3^{-16} \cdot 5^{-3} \cdot 7^{-1} \cdot 11^{-4} \cdot 13^{3} \cdot 59^{3} \cdot 743^{3}$ | 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: | $2.9876963440371360316933865690\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.5154768544539158016888728531\dots$ | 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: | $1.3279495871643759595575203249\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.10474191830635558892932495150\dots$ | 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: | $ 192 $ = $ 2^{4}\cdot3\cdot1\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)
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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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 6.6764153843390254618243893879 $ | 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
$\displaystyle 6.676415384 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.104742 \cdot 1.327950 \cdot 192}{2^2} \approx 6.676415384$
Modular invariants
Modular form 416955.2.a.ba
For more coefficients, see the Downloads section to the right.
Modular degree: | 31850496 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$19$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 58520 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 58513 & 8 \\ 58512 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 27719 & 0 \\ 0 & 58519 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 42561 & 46208 \\ 47044 & 9273 \end{array}\right),\left(\begin{array}{rr} 2699 & 2698 \\ 1938 & 37355 \end{array}\right),\left(\begin{array}{rr} 41896 & 24643 \\ 21565 & 27722 \end{array}\right),\left(\begin{array}{rr} 25803 & 35036 \\ 22762 & 58141 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 58514 & 58515 \end{array}\right),\left(\begin{array}{rr} 26848 & 24643 \\ 4845 & 27722 \end{array}\right)$.
The torsion field $K:=\Q(E[58520])$ is a degree-$50325057699840000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/58520\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 416955.ba
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1155.k4, its twist by $-19$.
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.