Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-110544x+14137344\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-110544xz^2+14137344z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-143265051x+660021716790\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(194, -46\right)\) |
$\hat{h}(P)$ | ≈ | $2.4705510938487420467795592583$ |
Torsion generators
\( \left(-384, 192\right) \), \( \left(204, 192\right) \)
Integral points
\( \left(-384, 192\right) \), \( \left(-102, 4986\right) \), \( \left(-102, -4884\right) \), \( \left(180, 192\right) \), \( \left(180, -372\right) \), \( \left(192, -96\right) \), \( \left(194, -46\right) \), \( \left(194, -148\right) \), \( \left(204, 192\right) \), \( \left(204, -396\right) \), \( \left(264, 1704\right) \), \( \left(264, -1968\right) \), \( \left(792, 20184\right) \), \( \left(792, -20976\right) \)
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: | $109979783424 $ | = | $2^{8} \cdot 3^{4} \cdot 7^{4} \cdot 47^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{149392736901862384897}{109979783424} \) | = | $2^{-8} \cdot 3^{-4} \cdot 7^{-4} \cdot 47^{-2} \cdot 5306113^{3}$ | 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: | $1.4291897166110780203567652087\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.4291897166110780203567652087\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0033341828506757\dots$ | |||
Szpiro ratio: | $6.122065013360366\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.4705510938487420467795592583\dots$ | 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: | $ 128 $ = $ 2^{3}\cdot2^{2}\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: | $8$ | 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
$\displaystyle 4.330017508 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.876326 \cdot 2.470551 \cdot 128}{8^2} \approx 4.330017508$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 8192 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$47$ | $2$ | $I_{2}$ | 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 | 8.48.0.24 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1128 = 2^{3} \cdot 3 \cdot 47 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 1124 & 1125 \end{array}\right),\left(\begin{array}{rr} 1121 & 8 \\ 1120 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 377 & 8 \\ 380 & 33 \end{array}\right),\left(\begin{array}{rr} 569 & 288 \\ 276 & 557 \end{array}\right),\left(\begin{array}{rr} 223 & 2 \\ 1038 & 1123 \end{array}\right),\left(\begin{array}{rr} 5 & 852 \\ 1122 & 275 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1128])$ is a degree-$1833099264$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1128\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{47})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-47})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.101185065216.5 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.4.15837088813056.11 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.2.33207591377710512.15 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.