Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+94468531x-1561262960140\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+94468531xz^2-1561262960140z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+1511496501x-99919317952442\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{152551433414953542560675222376054525931048330243}{35162954734447106086719846418064815529956}, \frac{59569125148074051080324003993795388477583497209960220540294240203748543}{6593682671467319075302676480383923096912931631704437042431896}\right)\) |
$\hat{h}(P)$ | ≈ | $108.64541431405362233526453277$ |
Torsion generators
\( \left(\frac{35787}{4}, -\frac{35791}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 52983 \) | = | $3^{2} \cdot 7 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-1106942513120216798702642847 $ | = | $-1 \cdot 3^{12} \cdot 7 \cdot 29^{14} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{215015459663151503}{2552757445339983} \) | = | $3^{-6} \cdot 7^{-1} \cdot 29^{-8} \cdot 599087^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $3.8701894328392064482548721503\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.6372353735119145889656135157\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0258598043382023\dots$ | |||
Szpiro ratio: | $6.405396238850258\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $108.64541431405362233526453277\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.024065826136045248071829874693\dots$ | 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: | $ 8 $ = $ 2\cdot1\cdot2^{2} $ | 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)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 5.2292833027212323440247026835 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 5.229283303 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.024066 \cdot 108.645414 \cdot 8}{2^2} \approx 5.229283303$
Modular invariants
Modular form 52983.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 20643840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$29$ | $4$ | $I_{8}^{*}$ | Additive | 1 | 2 | 14 | 8 |
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 | 16.24.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9744 = 2^{4} \cdot 3 \cdot 7 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9729 & 16 \\ 9728 & 17 \end{array}\right),\left(\begin{array}{rr} 6968 & 1 \\ 4255 & 10 \end{array}\right),\left(\begin{array}{rr} 5711 & 9728 \\ 6712 & 9615 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 2180 & 2121 \end{array}\right),\left(\begin{array}{rr} 6103 & 16 \\ 8270 & 2121 \end{array}\right),\left(\begin{array}{rr} 3235 & 9728 \\ 6752 & 315 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9740 & 9741 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 9646 & 9731 \end{array}\right)$.
The torsion field $K:=\Q(E[9744])$ is a degree-$8448450232320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9744\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 52983.f
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 609.a6, its twist by $-87$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{87}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-609}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{87})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{42}, \sqrt{87})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-6}, \sqrt{-58})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.107841329470224.18 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.9014654806327296.63 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | add | ord | split | ord | ord | ord | ord | ss | add | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1,1 | - | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 1 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0 | 0 |
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.