Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2+2699x+66385\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z+2699xz^2+66385z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+3497229x+3044796750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(9, 298\right)\) |
$\hat{h}(P)$ | ≈ | $1.8645757077907874810971620971$ |
Torsion generators
\( \left(42, 485\right) \)
Integral points
\( \left(9, 298\right) \), \( \left(9, -307\right) \), \( \left(42, 485\right) \), \( \left(42, -527\right) \), \( \left(111, 1267\right) \), \( \left(111, -1378\right) \), \( \left(548, 12629\right) \), \( \left(548, -13177\right) \)
Invariants
Conductor: | \( 10626 \) | = | $2 \cdot 3 \cdot 7 \cdot 11 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-3097446973236 $ | = | $-1 \cdot 2^{2} \cdot 3^{3} \cdot 7 \cdot 11^{4} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{2173106048486183}{3097446973236} \) | = | $2^{-2} \cdot 3^{-3} \cdot 7^{-1} \cdot 11^{-4} \cdot 23^{-4} \cdot 129527^{3}$ | 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: | $1.0823698491614812992726162164\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.0823698491614812992726162164\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0152719889942219\dots$ | |||
Szpiro ratio: | $3.850200011571723\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.8645757077907874810971620971\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.54098852899717056672418945966\dots$ | 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: | $ 32 $ = $ 2\cdot1\cdot1\cdot2^{2}\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: | $4$ | 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) $ ≈ $ 2.0174281387231925331949555686 $ | 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
$\displaystyle 2.017428139 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.540989 \cdot 1.864576 \cdot 32}{4^2} \approx 2.017428139$
Modular invariants
Modular form 10626.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 24576 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$11$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$23$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 42504 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 30364 & 1 \\ 36455 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 42498 & 42499 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 26568 & 5321 \\ 26593 & 26640 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22177 & 8 \\ 3700 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 42497 & 8 \\ 42496 & 9 \end{array}\right),\left(\begin{array}{rr} 28340 & 1 \\ 14191 & 6 \end{array}\right),\left(\begin{array}{rr} 37195 & 37194 \\ 26578 & 5323 \end{array}\right),\left(\begin{array}{rr} 15457 & 8 \\ 19324 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[42504])$ is a degree-$10920439421337600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42504\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 10626a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-21}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | 4.2.86028096.2 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.351298031616.4 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | nonsplit | nonsplit | ord | nonsplit | split | ord | ord | ord | split | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 1 | 1 | 3 | 6 | 3 | 1 | 1 | 2 | 1 | 1,1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 2 | 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.