Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-188611873x+1437325066847\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-188611873xz^2+1437325066847z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-15277561740x+1047855806416656\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(7466, 667359\right)\) |
$\hat{h}(P)$ | ≈ | $7.8286625022904740759672020790$ |
Torsion generators
\( \left(-16591, 0\right) \)
Integral points
\( \left(-16591, 0\right) \), \((7466,\pm 667359)\)
Invariants
Conductor: | \( 466752 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-463123585349116788549353472 $ | = | $-1 \cdot 2^{30} \cdot 3^{2} \cdot 11^{2} \cdot 13^{12} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{2830680648734534916567625}{1766676274677722124288} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-12} \cdot 17^{-1} \cdot 4041683^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.8183601806390668831297250136\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.7786394097991489190038768314\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $7.8286625022904740759672020790\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.048706391127381380238059224081\dots$ | 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);
|
Tamagawa product: | $ 16 $ = $ 2\cdot2\cdot2\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)
|
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: | $9$ = $3^2$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 13.727012322269666191648367418 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 13.727012322 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.048706 \cdot 7.828663 \cdot 16}{2^2} \approx 13.727012322$
Modular invariants
Modular form 466752.2.a.dt
For more coefficients, see the Downloads section to the right.
Modular degree: | 175177728 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{20}^{*}$ | Additive | -1 | 6 | 30 | 12 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$13$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$17$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2058 & 1321 \\ 2057 & 1310 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4438 & 4479 \end{array}\right),\left(\begin{array}{rr} 409 & 12 \\ 210 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2894 & 4485 \\ 3855 & 2252 \end{array}\right),\left(\begin{array}{rr} 3739 & 4476 \\ 3734 & 4415 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 4477 & 12 \\ 4476 & 13 \end{array}\right),\left(\begin{array}{rr} 4477 & 4486 \\ 2294 & 2253 \end{array}\right),\left(\begin{array}{rr} 2243 & 0 \\ 0 & 4487 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2255 & 4476 \\ 2256 & 4475 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$794139033600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 466752.dt
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 14586.e2, its twist by $-8$.
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.