Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-6864x+223488\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-6864xz^2+223488z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-8895771x+10453743414\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{8}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(54, 78\right)\) |
$\hat{h}(P)$ | ≈ | $1.2352755469243710233897796292$ |
Torsion generators
\( \left(12, 372\right) \)
Integral points
\( \left(-96, 48\right) \), \( \left(-72, 624\right) \), \( \left(-72, -552\right) \), \( \left(-24, 624\right) \), \( \left(-24, -600\right) \), \( \left(12, 372\right) \), \( \left(12, -384\right) \), \( \left(32, 176\right) \), \( \left(32, -208\right) \), \( \left(48, 48\right) \), \( \left(48, -96\right) \), \( \left(54, 78\right) \), \( \left(54, -132\right) \), \( \left(66, 210\right) \), \( \left(66, -276\right) \), \( \left(96, 624\right) \), \( \left(96, -720\right) \), \( \left(192, 2352\right) \), \( \left(192, -2544\right) \), \( \left(768, 20784\right) \), \( \left(768, -21552\right) \), \( \left(729536, 622753040\right) \), \( \left(729536, -623482576\right) \)
Invariants
Conductor: | \( 1974 \) | = | $2 \cdot 3 \cdot 7 \cdot 47$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-990247845888 $ | = | $-1 \cdot 2^{16} \cdot 3^{8} \cdot 7^{2} \cdot 47 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{35765103905346817}{990247845888} \) | = | $-1 \cdot 2^{-16} \cdot 3^{-8} \cdot 7^{-2} \cdot 47^{-1} \cdot 329473^{3}$ | 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: | $1.0826161263311053656481491480\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.0826161263311053656481491480\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9692229522993849\dots$ | |||
Szpiro ratio: | $5.029441635943601\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.2352755469243710233897796292\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.87632624137394673930805623341\dots$ | 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: | $ 256 $ = $ 2^{4}\cdot2^{3}\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)
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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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 4.3300175083895217325433320651 $ | 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
$\displaystyle 4.330017508 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.876326 \cdot 1.235276 \cdot 256}{8^2} \approx 4.330017508$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4096 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$3$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$47$ | $1$ | $I_{1}$ | Non-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 | 8.48.0.159 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2256 = 2^{4} \cdot 3 \cdot 47 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 2252 & 2253 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1985 & 16 \\ 1610 & 1983 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 572 & 693 \end{array}\right),\left(\begin{array}{rr} 2241 & 16 \\ 2240 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2158 & 2243 \end{array}\right),\left(\begin{array}{rr} 1544 & 1 \\ 1711 & 10 \end{array}\right),\left(\begin{array}{rr} 1505 & 16 \\ 760 & 129 \end{array}\right)$.
The torsion field $K:=\Q(E[2256])$ is a degree-$29329588224$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2256\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 1974.h
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-47}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | 4.2.331632.4 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.6625509377261824.4 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.873116441649.1 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.0.242945341583616.30 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.2.33207591377710512.15 | \(\Z/24\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | 5 | 4 | 3 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.