Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2+1683x+176341\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z+1683xz^2+176341z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+26925x+11312750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(89, 968\right)\) | \(\left(35, 509\right)\) |
$\hat{h}(P)$ | ≈ | $0.71764855958391983400342042678$ | $2.4067791361502805773208037453$ |
Torsion generators
\( \left(-46, 23\right) \)
Integral points
\( \left(-46, 23\right) \), \( \left(-21, 373\right) \), \( \left(-21, -352\right) \), \( \left(14, 443\right) \), \( \left(14, -457\right) \), \( \left(35, 509\right) \), \( \left(35, -544\right) \), \( \left(54, 623\right) \), \( \left(54, -677\right) \), \( \left(89, 968\right) \), \( \left(89, -1057\right) \), \( \left(150, 1871\right) \), \( \left(150, -2021\right) \), \( \left(278, 4559\right) \), \( \left(278, -4837\right) \), \( \left(689, 17768\right) \), \( \left(689, -18457\right) \), \( \left(2114, 96143\right) \), \( \left(2114, -98257\right) \)
Invariants
Conductor: | \( 84150 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-13802703750000 $ | = | $-1 \cdot 2^{4} \cdot 3^{10} \cdot 5^{7} \cdot 11 \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{46268279}{1211760} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-1} \cdot 11^{-1} \cdot 17^{-1} \cdot 359^{3}$ | 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: | $1.2014620429606458505438489209\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.15256305759045918245415336417\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8504769920867886\dots$ | |||
Szpiro ratio: | $3.3233754425229094\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.6960346943149748662380966544\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.53004424033042337780717569518\dots$ | 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: | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1\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! $ ≈ $ 7.1917873689777814893439062452 $ | 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
$\displaystyle 7.191787369 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.530044 \cdot 1.696035 \cdot 32}{2^2} \approx 7.191787369$
Modular invariants
Modular form 84150.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 196608 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$11$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$17$ | $1$ | $I_{1}$ | Non-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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 14033 & 14028 \\ 14030 & 2807 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13204 & 1 \\ 19823 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 8972 & 22439 \\ 17929 & 22434 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 19643 & 19638 \\ 2810 & 14027 \end{array}\right),\left(\begin{array}{rr} 14959 & 22432 \\ 14956 & 22407 \end{array}\right),\left(\begin{array}{rr} 12248 & 3 \\ 2045 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$762373472256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 84150.e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 5610.v4, its twist by $-15$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-935}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{165}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-51}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-51}, \sqrt{165})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | add | ord | nonsplit | ord | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 7 | - | - | 2 | 2 | 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 |
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.