Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-21675x+984406\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-21675xz^2+984406z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-346800x+63002000\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-85, 1487\right)\) | \(\left(34, 535\right)\) |
$\hat{h}(P)$ | ≈ | $0.44184182828348666314712458216$ | $0.90547199361790063447606231745$ |
Integral points
\( \left(-155, 787\right) \), \( \left(-155, -788\right) \), \( \left(-85, 1487\right) \), \( \left(-85, -1488\right) \), \( \left(34, 535\right) \), \( \left(34, -536\right) \), \( \left(115, 112\right) \), \( \left(115, -113\right) \), \( \left(170, 1487\right) \), \( \left(170, -1488\right) \), \( \left(221, 2643\right) \), \( \left(221, -2644\right) \), \( \left(340, 5737\right) \), \( \left(340, -5738\right) \), \( \left(1334, 48435\right) \), \( \left(1334, -48436\right) \), \( \left(6170, 484512\right) \), \( \left(6170, -484513\right) \)
Invariants
Conductor: | \( 455175 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $233082315703125 $ | = | $3^{6} \cdot 5^{7} \cdot 7^{2} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1183744}{245} \) | = | $2^{12} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{2}$ | 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: | $1.4720418364482933441041793910\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.82638771212155038231033443336\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7633283575703148\dots$ | |||
Szpiro ratio: | $3.1903555794135228\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.35305662388889676735737811909\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.52755528672567372132190561032\dots$ | 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: | $ 48 $ = $ 2\cdot2^{2}\cdot2\cdot3 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 8.9403306454130534602800572913 $ | 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
$\displaystyle 8.940330645 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.527555 \cdot 0.353057 \cdot 48}{1^2} \approx 8.940330645$
Modular invariants
Modular form 455175.2.a.u
For more coefficients, see the Downloads section to the right.
Modular degree: | 2239488 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$17$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
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 label 10.2.0.a.1, level \( 10 = 2 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 2 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 9 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[10])$ is a degree-$1440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 455175u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 10115d1, its twist by $-15$.
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.