Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-544326x+154522980\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-544326xz^2+154522980z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-705446523x+7211540494422\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{6}\Z\)
Torsion generators
\( \left(428, -214\right) \), \( \left(408, 426\right) \)
Integral points
\( \left(-852, 426\right) \), \( \left(156, 8490\right) \), \( \left(156, -8646\right) \), \( \left(408, 426\right) \), \( \left(408, -834\right) \), \( \left(428, -214\right) \), \( \left(444, 426\right) \), \( \left(444, -870\right) \), \( \left(768, 13386\right) \), \( \left(768, -14154\right) \)
Invariants
Conductor: | \( 3570 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $770635366502400 $ | = | $2^{12} \cdot 3^{12} \cdot 5^{2} \cdot 7^{2} \cdot 17^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{17836145204788591940449}{770635366502400} \) | = | $2^{-12} \cdot 3^{-12} \cdot 5^{-2} \cdot 7^{-2} \cdot 17^{-2} \cdot 73^{3} \cdot 357913^{3}$ | 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: | $1.9353374397635033655273729871\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.9353374397635033655273729871\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0070556832365427\dots$ | |||
Szpiro ratio: | $6.263264579108794\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.47436358575516484126690720625\dots$ | 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: | $ 1152 $ = $ ( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot2\cdot2\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: | $12$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 3.7949086860413187301352576500 $ | 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
$\displaystyle 3.794908686 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.474364 \cdot 1.000000 \cdot 1152}{12^2} \approx 3.794908686$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 36864 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7140 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 7124 & 7133 \end{array}\right),\left(\begin{array}{rr} 1261 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7129 & 12 \\ 7128 & 13 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 6114 & 7135 \end{array}\right),\left(\begin{array}{rr} 4761 & 4 \\ 4768 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2857 & 12 \\ 2862 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3577 & 12 \\ 7074 & 7027 \end{array}\right)$.
The torsion field $K:=\Q(E[7140])$ is a degree-$909650165760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7140\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 3570v
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{7}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.3384009916875.2 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.115097703310973880000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 17 |
---|---|---|---|---|---|
Reduction type | split | split | nonsplit | split | nonsplit |
$\lambda$-invariant(s) | 6 | 1 | 0 | 1 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.