Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-187273275x+986319532125\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-187273275xz^2+986319532125z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-242706165075x+46021364683296750\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 417450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $19034065900929024000000 $ | = | $2^{25} \cdot 3^{4} \cdot 5^{6} \cdot 11^{7} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{26240674555395219529}{687630974976} \) | = | $2^{-25} \cdot 3^{-4} \cdot 11^{-1} \cdot 23^{-1} \cdot 983^{3} \cdot 3023^{3}$ | 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: | $3.3810009318435813889588831391\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3773343392273459296275316835\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0028524391908074\dots$ | |||
Szpiro ratio: | $5.312800663322391\dots$ |
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: | $0.11339100628369476739679229192\dots$ | 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: | $ 4 $ = $ 1\cdot2\cdot1\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) $ ≈ $ 0.45356402513477906958716916769 $ | 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
$\displaystyle 0.453564025 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.113391 \cdot 1.000000 \cdot 4}{1^2} \approx 0.453564025$
Modular invariants
Modular form 417450.2.a.i
For more coefficients, see the Downloads section to the right.
Modular degree: | 80640000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{25}$ | Non-split multiplicative | 1 | 1 | 25 | 25 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$5$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$11$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$23$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2024 = 2^{3} \cdot 11 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 2023 & 2 \\ 2022 & 3 \end{array}\right),\left(\begin{array}{rr} 1013 & 2 \\ 1013 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2023 & 0 \end{array}\right),\left(\begin{array}{rr} 1289 & 2 \\ 1289 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1519 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1937 & 2 \\ 1937 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[2024])$ is a degree-$2708442316800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2024\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 417450.i consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1518.h1, its twist by $-55$.
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 $0$.