Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-226250x-60187290\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-226250xz^2-60187290z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-293220027x-2807218542186\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 6690 \) | = | $2 \cdot 3 \cdot 5 \cdot 223$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-822726139895701410 $ | = | $-1 \cdot 2 \cdot 3 \cdot 5 \cdot 223^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1280824409818832580001}{822726139895701410} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{-1} \cdot 19^{3} \cdot 223^{-7} \cdot 571579^{3}$ | 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: | $2.1388987187680500150181604995\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.1388987187680500150181604995\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9932626136046446\dots$ | |||
Szpiro ratio: | $5.602451666346784\dots$ |
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: | $0.10637308166537897417814166136\dots$ | 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: | $ 1 $ | 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: | $1$ | 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: | $49$ = $7^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 5.2122810016035697347289414064 $ | 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
$\displaystyle 5.212281002 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{49 \cdot 0.106373 \cdot 1.000000 \cdot 1}{1^2} \approx 5.212281002$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 139944 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$223$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
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 |
---|---|---|
$7$ | 7B.1.3 | 7.48.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 187320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 223 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 28568 & 7 \\ 21833 & 187314 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 46823 & 187314 \end{array}\right),\left(\begin{array}{rr} 46833 & 133808 \\ 93646 & 153833 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 187307 & 14 \\ 187306 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 124888 & 7 \\ 124873 & 187314 \end{array}\right),\left(\begin{array}{rr} 74936 & 7 \\ 74921 & 187314 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 93653 & 187314 \end{array}\right)$.
The torsion field $K:=\Q(E[187320])$ is a degree-$1829581732781752320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/187320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 6690j
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.26760.3 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.19162771776000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | Not in database |
$7$ | 7.1.600362847000000.17 | \(\Z/7\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$18$ | 18.0.1743360712783446466704197220691968000000.1 | \(\Z/14\Z\) | Not in database |
$21$ | 21.1.2916867225003158090761647087922058466205473669120000000000000000000.1 | \(\Z/14\Z\) | 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 | 223 |
---|---|---|---|---|---|
Reduction type | split | split | split | ord | nonsplit |
$\lambda$-invariant(s) | 4 | 1 | 1 | 4 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.