Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-643x+17153\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-643xz^2+17153z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-833355x+802790406\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(16, 97\right)\) |
$\hat{h}(P)$ | ≈ | $1.7872265459391410313734538185$ |
Torsion generators
\( \left(44, 251\right) \)
Integral points
\( \left(-34, 17\right) \), \( \left(-16, 161\right) \), \( \left(-16, -145\right) \), \( \left(2, 125\right) \), \( \left(2, -127\right) \), \( \left(16, 97\right) \), \( \left(16, -113\right) \), \( \left(44, 251\right) \), \( \left(44, -295\right) \), \( \left(122, 1265\right) \), \( \left(122, -1387\right) \), \( \left(254, 3905\right) \), \( \left(254, -4159\right) \)
Invariants
Conductor: | \( 22386 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 41$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-110884842432 $ | = | $-1 \cdot 2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 13^{2} \cdot 41 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{29403487464625}{110884842432} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 13^{-2} \cdot 41^{-1} \cdot 6173^{3}$ | 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: | $0.80493670184596441062838437328\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.80493670184596441062838437328\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.7872265459391410313734538185\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.92179379912164921937547511426\dots$ | 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: | $ 216 $ = $ ( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3\cdot2\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: | $6$ | 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'(E,1) $ ≈ $ 9.8847260860338212896028244157 $ | 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
$\displaystyle 9.884726086 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.921794 \cdot 1.787227 \cdot 216}{6^2} \approx 9.884726086$
Modular invariants
Modular form 22386.2.a.ba
For more coefficients, see the Downloads section to the right.
Modular degree: | 20736 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$13$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$41$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 89544 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 41 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 41329 & 12 \\ 68886 & 73 \end{array}\right),\left(\begin{array}{rr} 25594 & 3 \\ 38349 & 89536 \end{array}\right),\left(\begin{array}{rr} 44773 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 59705 & 2 \\ 14928 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3735 & 18658 \\ 70910 & 85829 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 89494 & 89535 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 89533 & 12 \\ 89532 & 13 \end{array}\right),\left(\begin{array}{rr} 26218 & 3 \\ 54573 & 89536 \end{array}\right)$.
The torsion field $K:=\Q(E[89544])$ is a degree-$111799215901900800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/89544\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 22386ba
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-287}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | 4.2.27937728.6 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.2179077117867.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$9$ | 9.3.8719586111522478583564992.1 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.3021378577354095953535508554809452474046882138963414581604352.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | split | ss | split | ss | split | ord | ord | ss | ord | ord | ord | nonsplit | ord | ss |
$\lambda$-invariant(s) | 4 | 2 | 1,1 | 4 | 1,1 | 2 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3,1 |
$\mu$-invariant(s) | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.