Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-1825650x+944824500\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1825650xz^2+944824500z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2366043075x+44117222514750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(116, 27046\right)\) | \(\left(721, 1636\right)\) |
$\hat{h}(P)$ | ≈ | $2.5234959142228934911558481085$ | $3.7351712525465114675381053973$ |
Torsion generators
\( \left(820, -410\right) \)
Integral points
\( \left(116, 27046\right) \), \( \left(116, -27162\right) \), \( \left(545, 10315\right) \), \( \left(545, -10860\right) \), \( \left(721, 1636\right) \), \( \left(721, -2357\right) \), \( \left(820, -410\right) \), \( \left(884, 4326\right) \), \( \left(884, -5210\right) \), \( \left(3845, 223440\right) \), \( \left(3845, -227285\right) \)
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: | $3168401417280000000 $ | = | $2^{12} \cdot 3^{5} \cdot 5^{7} \cdot 11^{6} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{24310870577209}{114462720} \) | = | $2^{-12} \cdot 3^{-5} \cdot 5^{-1} \cdot 23^{-1} \cdot 59^{3} \cdot 491^{3}$ | 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: | $2.3996602053965547852080179290\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.39599361278031932587666647340\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9566815063794372\dots$ | |||
Szpiro ratio: | $4.239397522986632\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $8.3904949878078709725023785000\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.25355572580198821793622275646\dots$ | 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: | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2}\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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 8.5098321858862760320565566808 $ | 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
$\displaystyle 8.509832186 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.253556 \cdot 8.390495 \cdot 16}{2^2} \approx 8.509832186$
Modular invariants
Modular form 417450.2.a.z
For more coefficients, see the Downloads section to the right.
Modular degree: | 14745600 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$3$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
$5$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$23$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ 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 \( 30360 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 30353 & 8 \\ 30352 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 30354 & 30355 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 4489 & 4488 \\ 16918 & 21055 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9932 & 27599 \\ 3289 & 16554 \end{array}\right),\left(\begin{array}{rr} 23123 & 12078 \\ 10010 & 14147 \end{array}\right),\left(\begin{array}{rr} 2759 & 0 \\ 0 & 30359 \end{array}\right),\left(\begin{array}{rr} 24124 & 2761 \\ 2783 & 13806 \end{array}\right),\left(\begin{array}{rr} 9208 & 8283 \\ 3685 & 2762 \end{array}\right)$.
The torsion field $K:=\Q(E[30360])$ is a degree-$2600104624128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 417450.z
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 690.e3, its twist by $-55$.
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.