Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-71x-196\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-71xz^2-196z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-92043x-7755642\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-4, 8\right)\) | \(\left(22, 86\right)\) |
$\hat{h}(P)$ | ≈ | $0.72234430467980239904282360010$ | $2.0134699422604012392141562378$ |
Torsion generators
\( \left(-3, 1\right) \), \( \left(9, -5\right) \)
Integral points
\( \left(-7, 7\right) \), \( \left(-7, -1\right) \), \( \left(-6, 10\right) \), \( \left(-6, -5\right) \), \( \left(-4, 8\right) \), \( \left(-4, -5\right) \), \( \left(-3, 1\right) \), \( \left(9, -5\right) \), \( \left(13, 29\right) \), \( \left(13, -43\right) \), \( \left(14, 35\right) \), \( \left(14, -50\right) \), \( \left(22, 86\right) \), \( \left(22, -109\right) \), \( \left(24, 100\right) \), \( \left(24, -125\right) \), \( \left(48, 307\right) \), \( \left(48, -356\right) \), \( \left(144, 1660\right) \), \( \left(144, -1805\right) \), \( \left(201, 2755\right) \), \( \left(201, -2957\right) \), \( \left(269, 4285\right) \), \( \left(269, -4555\right) \), \( \left(6134, 477395\right) \), \( \left(6134, -483530\right) \)
Invariants
Conductor: | \( 3315 \) | = | $3 \cdot 5 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $10989225 $ | = | $3^{2} \cdot 5^{2} \cdot 13^{2} \cdot 17^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{39616946929}{10989225} \) | = | $3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 13^{-2} \cdot 17^{-2} \cdot 487^{3}$ | 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: | $0.058381471282608958966053860809\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $0.058381471282608958966053860809\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8470603652891601\dots$ | |||
Szpiro ratio: | $3.010348125513297\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.3563434593337876454091229803\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.6789888805069973421081518821\dots$ | 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: | $ 16 $ = $ 2\cdot2\cdot2\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: | $4$ | 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! $ ≈ $ 2.2772855863698241937913070595 $ | 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
$\displaystyle 2.277285586 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.678989 \cdot 1.356343 \cdot 16}{4^2} \approx 2.277285586$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 896 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$13$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$17$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13260 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 13257 & 4 \\ 13256 & 5 \end{array}\right),\left(\begin{array}{rr} 6633 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10923 & 2 \\ 1558 & 13259 \end{array}\right),\left(\begin{array}{rr} 7957 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 11221 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4421 & 4 \\ 8842 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[13260])$ is a degree-$94603617239040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13260\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 3315.a
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{-17}, \sqrt{-65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | nonsplit | ord | ord | split | nonsplit | ord | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 2 | 2 | 2 | 2 | 3 | 2 | 2 | 2,2 | 2 | 2,2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 1 | 0 | 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.