Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-39759x+3047929\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-39759xz^2+3047929z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-51527691x+142358758470\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(114, -71\right)\) | \(\left(110, 37\right)\) |
$\hat{h}(P)$ | ≈ | $0.48809364641140446676320835853$ | $0.64995493919412363489523763842$ |
Torsion generators
\( \left(\frac{463}{4}, -\frac{463}{8}\right) \)
Integral points
\( \left(-180, 2183\right) \), \( \left(-180, -2003\right) \), \( \left(-166, 2337\right) \), \( \left(-166, -2171\right) \), \( \left(-26, 2029\right) \), \( \left(-26, -2003\right) \), \( \left(30, 1357\right) \), \( \left(30, -1387\right) \), \( \left(100, 223\right) \), \( \left(100, -323\right) \), \( \left(110, 37\right) \), \( \left(110, -147\right) \), \( \left(114, -43\right) \), \( \left(114, -71\right) \), \( \left(122, 69\right) \), \( \left(122, -191\right) \), \( \left(128, 181\right) \), \( \left(128, -309\right) \), \( \left(156, 727\right) \), \( \left(156, -883\right) \), \( \left(206, 1797\right) \), \( \left(206, -2003\right) \), \( \left(422, 7629\right) \), \( \left(422, -8051\right) \), \( \left(478, 9421\right) \), \( \left(478, -9899\right) \), \( \left(1122, 36469\right) \), \( \left(1122, -37591\right) \), \( \left(3656, 218917\right) \), \( \left(3656, -222573\right) \), \( \left(96756, 30048157\right) \), \( \left(96756, -30144913\right) \)
Invariants
Conductor: | \( 9338 \) | = | $2 \cdot 7 \cdot 23 \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $462042447104 $ | = | $2^{8} \cdot 7^{6} \cdot 23^{2} \cdot 29 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{6950735348004218737}{462042447104} \) | = | $2^{-8} \cdot 7^{-6} \cdot 23^{-2} \cdot 29^{-1} \cdot 1908433^{3}$ | 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: | $1.2940259741488626717552595338\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.2940259741488626717552595338\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9351328122549867\dots$ | |||
Szpiro ratio: | $4.7457999845267915\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.26708467404471710979841645853\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.88928495400691342191604565148\dots$ | 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: | $ 96 $ = $ 2^{3}\cdot( 2 \cdot 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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 5.7003451697713852312578812650 $ | 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
$\displaystyle 5.700345170 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.889285 \cdot 0.267085 \cdot 96}{2^2} \approx 5.700345170$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 36864 | 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 |
$7$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$23$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$29$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18676 = 2^{2} \cdot 7 \cdot 23 \cdot 29 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 5337 & 4 \\ 10674 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14009 & 4670 \\ 4668 & 14007 \end{array}\right),\left(\begin{array}{rr} 6442 & 1 \\ 15455 & 0 \end{array}\right),\left(\begin{array}{rr} 13805 & 4 \\ 8934 & 9 \end{array}\right),\left(\begin{array}{rr} 18673 & 4 \\ 18672 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[18676])$ is a degree-$2939004624568320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18676\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 9338.f
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{29}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.4.12027344.1 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | ord | split | ord | ord | ord | ord | nonsplit | split | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 |
$\mu$-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.