Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-46366756x-121538372763\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-46366756xz^2-121538372763z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-60091316208x-5669773223826096\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{324419555248790961414321532035717}{8167677748751632240041796}, \frac{5843331645820759668797883467835503206817005177643}{23342528183922034476827759433066233656}\right)\) |
$\hat{h}(P)$ | ≈ | $70.515883950173463768076651064$ |
Integral points
None
Invariants
Conductor: | \( 5929 \) | = | $7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2292646180979 $ | = | $-1 \cdot 7^{6} \cdot 11^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{52893159101157376}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1} \cdot 29^{3} \cdot 809^{3}$ | 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: | $2.6686118260254890471950135659\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.49670911509864712261136540520\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0929566831983986\dots$ | |||
Szpiro ratio: | $7.432409680936172\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $70.515883950173463768076651064\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.028927964792726786844322657879\dots$ | 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: | $ 4 $ = $ 2\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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 8.1595240329545034357855133844 $ | 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
$\displaystyle 8.159524033 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.028928 \cdot 70.515884 \cdot 4}{1^2} \approx 8.159524033$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 216000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 |
---|---|---|
$5$ | 5B.4.2 | 25.60.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 38 & 41 \\ 1291 & 1089 \end{array}\right),\left(\begin{array}{rr} 2640 & 1659 \\ 2877 & 1863 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 3801 & 50 \\ 3800 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1649 & 0 \\ 0 & 3849 \end{array}\right),\left(\begin{array}{rr} 3289 & 2730 \\ 2485 & 3009 \end{array}\right)$.
The torsion field $K:=\Q(E[3850])$ is a degree-$39916800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3850\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 5929h
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a2, its twist by $77$.
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 |
---|---|---|---|
$3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.0.741125.2 | \(\Z/5\Z\) | Not in database |
$6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.9302443780707.2 | \(\Z/3\Z\) | Not in database |
$10$ | 10.2.387011980885126953125.1 | \(\Z/5\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/10\Z\) | Not in database |
$20$ | 20.0.4570870158344417423828021183726377785205841064453125.1 | \(\Z/25\Z\) | 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 | ss | ord | ord | add | add | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | ? | 1 | 3 | - | - | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | ? | 0 | 2 | - | - | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.