Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-32417175x-295794110500\)
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(homogenize, simplify) |
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\(y^2z=x^3-32417175xz^2-295794110500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-32417175x-295794110500\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{81610389704140817044}{2335478921664849}, \frac{724808803681304250357930805772}{112866206403160372999257}\right)\)
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| $\hat{h}(P)$ | ≈ | $44.026656780978631056504781339$ |
Torsion generators
\( \left(8260, 0\right) \)
Integral points
\( \left(8260, 0\right) \)
Invariants
| Conductor: | \( 705600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-35617229448486328125000000 $ | = | $-1 \cdot 2^{6} \cdot 3^{11} \cdot 5^{18} \cdot 7^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{43927191786304}{415283203125} \) | = | $-1 \cdot 2^{6} \cdot 3^{-5} \cdot 5^{-12} \cdot 7^{-1} \cdot 8821^{3}$ | 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: | $3.5852942217889388765317187831\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.91174045643020453627242406558\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0456675584451367\dots$ | |||
| Szpiro ratio: | $4.9268561090394245\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $44.026656780978631056504781339\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.027617106674373739013333763233\dots$ | 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: | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\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) $ ≈ $ 9.7271110146906142981415124629 $ | 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
$\displaystyle 9.727111015 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.027617 \cdot 44.026657 \cdot 32}{2^2} \approx 9.727111015$
Modular invariants
Modular form 705600.2.a.hu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 141557760 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(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$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | Additive | 1 | 6 | 6 | 0 |
| $3$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
| $5$ | $4$ | $I_{12}^{*}$ | Additive | 1 | 2 | 18 | 12 |
| $7$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 | 8.12.0.14 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 517 & 522 \\ 310 & 733 \end{array}\right),\left(\begin{array}{rr} 556 & 839 \\ 257 & 834 \end{array}\right),\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 528 & 113 \\ 97 & 84 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 592 & 837 \\ 475 & 838 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 834 & 835 \end{array}\right),\left(\begin{array}{rr} 503 & 832 \\ 332 & 807 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 4.
Its isogeny class 705600.hu
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3360.d4, its twist by $-840$.
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.