Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+40351x-7193185\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+40351xz^2-7193185z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+52294869x-336389654106\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(215, 3280\right)\) | \(\left(369, 7438\right)\) |
$\hat{h}(P)$ | ≈ | $0.77208528997015816025047916699$ | $0.82577579345169849058659669579$ |
Torsion generators
\( \left(127, -64\right) \)
Integral points
\( \left(127, -64\right) \), \( \left(131, 536\right) \), \( \left(131, -668\right) \), \( \left(171, 2092\right) \), \( \left(171, -2264\right) \), \( \left(215, 3280\right) \), \( \left(215, -3496\right) \), \( \left(271, 4736\right) \), \( \left(271, -5008\right) \), \( \left(369, 7438\right) \), \( \left(369, -7808\right) \), \( \left(495, 11344\right) \), \( \left(495, -11840\right) \), \( \left(527, 12416\right) \), \( \left(527, -12944\right) \), \( \left(1139, 38392\right) \), \( \left(1139, -39532\right) \), \( \left(1909, 82898\right) \), \( \left(1909, -84808\right) \), \( \left(2151, 99112\right) \), \( \left(2151, -101264\right) \), \( \left(4527, 302656\right) \), \( \left(4527, -307184\right) \), \( \left(4635, 313564\right) \), \( \left(4635, -318200\right) \), \( \left(15615, 1943680\right) \), \( \left(15615, -1959296\right) \), \( \left(21269, 3091442\right) \), \( \left(21269, -3112712\right) \), \( \left(22391, 3339536\right) \), \( \left(22391, -3361928\right) \), \( \left(574299, 434931592\right) \), \( \left(574299, -435505892\right) \)
Invariants
Conductor: | \( 116886 \) | = | $2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-26661886699650048 $ | = | $-1 \cdot 2^{10} \cdot 3^{4} \cdot 7^{3} \cdot 11^{6} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{4101378352343}{15049939968} \) | = | $2^{-10} \cdot 3^{-4} \cdot 7^{-3} \cdot 23^{-2} \cdot 16007^{3}$ | 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: | $1.8347100626292182971772239959\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $0.63576242623003302514625220692\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9679297576249781\dots$ | |||
Szpiro ratio: | $3.8653938012290405\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.59980767519172864189952632195\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.19097843163606858063578492553\dots$ | 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: | $ 480 $ = $ ( 2 \cdot 5 )\cdot2\cdot3\cdot2^{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)
|
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);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 13.746039490967133218566216766 $ | 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
$\displaystyle 13.746039491 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.190978 \cdot 0.599808 \cdot 480}{2^2} \approx 13.746039491$
Modular invariants
Modular form 116886.2.a.ba
For more coefficients, see the Downloads section to the right.
Modular degree: | 1382400 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$23$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1288 = 2^{3} \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 281 & 4 \\ 562 & 9 \end{array}\right),\left(\begin{array}{rr} 809 & 484 \\ 160 & 1127 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 370 & 1 \\ 1103 & 0 \end{array}\right),\left(\begin{array}{rr} 1285 & 4 \\ 1284 & 5 \end{array}\right),\left(\begin{array}{rr} 645 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1288])$ is a degree-$68942168064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1288\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 116886bg
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 966a1, its twist by $-11$.
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 |
$4$ | 4.2.28676032.13 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.40293425751986176.12 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | nonsplit | ord | split | add | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | 2 | 2 | 3 | - | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 0 | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.