Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+46321908x+144207134224\) | (homogenize, simplify) |
\(y^2z=x^3+46321908xz^2+144207134224z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+46321908x+144207134224\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(24258, 3942400\right)\) |
$\hat{h}(P)$ | ≈ | $2.6635520391047686434955639579$ |
Torsion generators
\( \left(-2692, 0\right) \)
Integral points
\( \left(-2692, 0\right) \), \((24258,\pm 3942400)\)
Invariants
Conductor: | \( 411840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-15344944902144000000000000 $ | = | $-1 \cdot 2^{24} \cdot 3^{9} \cdot 5^{12} \cdot 11^{4} \cdot 13 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{57519563401957999679}{80296734375000000} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-12} \cdot 11^{-4} \cdot 13^{-1} \cdot 23^{3} \cdot 157^{3} \cdot 1069^{3}$ | 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: | $3.5183794263656163789759156837\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.9293525111916435691524448831\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9984292783090328\dots$ | |||
Szpiro ratio: | $5.020901874600744\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.6635520391047686434955639579\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.047279189804781931199781315997\dots$ | 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: | $ 384 $ = $ 2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\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'(E,1) $ ≈ $ 12.089335911527836932379347921 $ | 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
$\displaystyle 12.089335912 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.047279 \cdot 2.663552 \cdot 384}{2^2} \approx 12.089335912$
Modular invariants
Modular form 411840.2.a.nt
For more coefficients, see the Downloads section to the right.
Modular degree: | 84934656 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{14}^{*}$ | Additive | -1 | 6 | 24 | 6 |
$3$ | $2$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$5$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$11$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$13$ | $1$ | $I_{1}$ | 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 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 7936 & 3 \\ 6141 & 17074 \end{array}\right),\left(\begin{array}{rr} 10297 & 24 \\ 3444 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 17144 & 8577 \\ 3319 & 5806 \end{array}\right),\left(\begin{array}{rr} 7801 & 24 \\ 7812 & 289 \end{array}\right),\left(\begin{array}{rr} 17137 & 24 \\ 17136 & 25 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 15854 & 7931 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9279 & 12152 \\ 14044 & 16531 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 4289 & 17136 \\ 4290 & 17159 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$31882346496000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 411840.nt
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 4290.bb8, its twist by $24$.
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.