Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-335944840x+1934592833600\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-335944840xz^2+1934592833600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-435384512667x+90261669397979574\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(6320, 249560\right)\) |
$\hat{h}(P)$ | ≈ | $2.4012592600501519239579672631$ |
Torsion generators
\( \left(14096, -7048\right) \), \( \left(3080, 962360\right) \)
Integral points
\( \left(-20720, 10360\right) \), \( \left(-19360, 1097000\right) \), \( \left(-19360, -1077640\right) \), \( \left(-14470, 1947860\right) \), \( \left(-14470, -1933390\right) \), \( \left(-9520, 2071160\right) \), \( \left(-9520, -2061640\right) \), \( \left(-4720, 1850360\right) \), \( \left(-4720, -1845640\right) \), \( \left(3080, 962360\right) \), \( \left(3080, -965440\right) \), \( \left(6320, 249560\right) \), \( \left(6320, -255880\right) \), \( \left(14096, -7048\right) \), \( \left(16310, 882980\right) \), \( \left(16310, -899290\right) \), \( \left(19280, 1610360\right) \), \( \left(19280, -1629640\right) \), \( \left(38780, 6852860\right) \), \( \left(38780, -6891640\right) \), \( \left(64280, 15650360\right) \), \( \left(64280, -15714640\right) \), \( \left(131600, 47229560\right) \), \( \left(131600, -47361160\right) \), \( \left(679280, 559310360\right) \), \( \left(679280, -559989640\right) \)
Invariants
Conductor: | \( 146370 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 41$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $809648781931584000000000000 $ | = | $2^{18} \cdot 3^{12} \cdot 5^{12} \cdot 7^{2} \cdot 17^{2} \cdot 41^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{4193026786024883569244771372161}{809648781931584000000000000} \) | = | $2^{-18} \cdot 3^{-12} \cdot 5^{-12} \cdot 7^{-2} \cdot 13^{3} \cdot 17^{-2} \cdot 37^{3} \cdot 41^{-2} \cdot 33524641^{3}$ | 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: | $3.8803084910858505943041887108\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $3.8803084910858505943041887108\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0008647107021558\dots$ | |||
Szpiro ratio: | $5.928334481670528\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $2.4012592600501519239579672631\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.047702609406234147985609360816\dots$ | 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: | $ 20736 $ = $ ( 2 \cdot 3^{2} )\cdot( 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)
|
Torsion order: | $12$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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) $ ≈ $ 16.494671889399632719419776634 $ | 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
$\displaystyle 16.494671889 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.047703 \cdot 2.401259 \cdot 20736}{12^2} \approx 16.494671889$
Modular invariants
Modular form 146370.2.a.cn
For more coefficients, see the Downloads section to the right.
Modular degree: | 69009408 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$5$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$41$ | $2$ | $I_{2}$ | 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 \( 117096 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \cdot 41 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 117080 & 117089 \end{array}\right),\left(\begin{array}{rr} 74257 & 12 \\ 94254 & 73 \end{array}\right),\left(\begin{array}{rr} 58555 & 6 \\ 117090 & 117091 \end{array}\right),\left(\begin{array}{rr} 110215 & 6 \\ 89538 & 117091 \end{array}\right),\left(\begin{array}{rr} 117085 & 12 \\ 117084 & 13 \end{array}\right),\left(\begin{array}{rr} 29281 & 6 \\ 117048 & 117055 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 39033 & 4 \\ 97588 & 33 \end{array}\right),\left(\begin{array}{rr} 100369 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[117096])$ is a degree-$83542271223398400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/117096\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 146370.cn
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 |
---|---|---|---|
$4$ | \(\Q(\sqrt{-7}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{34}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{-41})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.15299845194749787.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.127046328595984711766671875.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | split | split | ss | ord | nonsplit | ord | ss | ord | ord | ord | nonsplit | ord | ss |
$\lambda$-invariant(s) | 8 | 2 | 2 | 2 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.