Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-39x+90\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-39xz^2+90z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-50571x+4350726\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{8}\Z\)
Torsion generators
\( \left(3, 0\right) \)
Integral points
\( \left(-3, 15\right) \), \( \left(-3, -12\right) \), \( \left(3, 0\right) \), \( \left(3, -3\right) \), \( \left(6, 6\right) \), \( \left(6, -12\right) \)
Invariants
Conductor: | \( 21 \) | = | $3 \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $45927 $ | = | $3^{8} \cdot 7 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{6570725617}{45927} \) | = | $3^{-8} \cdot 7^{-1} \cdot 1873^{3}$ | 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: | $-0.27236820688903288395035020606\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.27236820688903288395035020606\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0015974661127929\dots$ | |||
Szpiro ratio: | $7.42510215304443\dots$ |
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: | $3.6089232431079364447472012512\dots$ | 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: | $ 8 $ = $ 2^{3}\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: | $8$ | 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) $ ≈ $ 0.45111540538849205559340015640 $ | 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
$\displaystyle 0.451115405 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.608923 \cdot 1.000000 \cdot 8}{8^2} \approx 0.451115405$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2 | 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 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $1$ | $I_{1}$ | nonsplit 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 | 8.48.0.159 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 336 = 2^{4} \cdot 3 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 332 & 333 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 238 & 323 \end{array}\right),\left(\begin{array}{rr} 321 & 16 \\ 320 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 92 & 213 \end{array}\right),\left(\begin{array}{rr} 218 & 263 \\ 43 & 54 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 113 & 16 \\ 232 & 129 \end{array}\right),\left(\begin{array}{rr} 248 & 1 \\ 223 & 10 \end{array}\right)$.
The torsion field $K:=\Q(E[336])$ is a degree-$12386304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/336\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | good | $2$ | \( 7 \) |
$3$ | split multiplicative | $4$ | \( 7 \) |
$7$ | nonsplit multiplicative | $8$ | \( 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 21.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | 2.2.28.1-63.1-a5 |
$4$ | 4.0.1008.1 | \(\Z/16\Z\) | not in database |
$8$ | 8.0.7710244864.3 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.796594176.14 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$8$ | 8.8.624529833984.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$8$ | 8.2.425329947.2 | \(\Z/24\Z\) | not in database |
$16$ | deg 16 | \(\Z/32\Z\) | 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 | 7 |
---|---|---|---|
Reduction type | ord | split | nonsplit |
$\lambda$-invariant(s) | 1 | 1 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.