Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+726x-333\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+726xz^2-333z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+11616x-21296\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(5, 58\right)\) |
$\hat{h}(P)$ | ≈ | $2.2791684848179374975941091120$ |
Integral points
\( \left(5, 58\right) \), \( \left(5, -59\right) \)
Invariants
Conductor: | \( 20691 \) | = | $3^{2} \cdot 11^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-24537891411 $ | = | $-1 \cdot 3^{6} \cdot 11^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{32768}{19} \) | = | $2^{15} \cdot 19^{-1}$ | 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: | $0.68308064905644822985344405411\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0651731316767918878751503533\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.3175706029138485\dots$ | |||
Szpiro ratio: | $3.157373383044805\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.2791684848179374975941091120\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.71011136120576866073841045556\dots$ | 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: | $ 2 $ = $ 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: | $1$ | 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) $ ≈ $ 3.2369268703427097606953324583 $ | 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
$\displaystyle 3.236926870 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.710111 \cdot 2.279168 \cdot 2}{1^2} \approx 3.236926870$
Modular invariants
Modular form 20691.2.a.i
For more coefficients, see the Downloads section to the right.
Modular degree: | 10800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11286 = 2 \cdot 3^{3} \cdot 11 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 7181 & 0 \\ 0 & 11285 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 5464 & 4525 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9085 & 6501 \\ 5885 & 5708 \end{array}\right),\left(\begin{array}{rr} 9769 & 10296 \\ 7007 & 8053 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 11233 & 54 \\ 11232 & 55 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[11286])$ is a degree-$2369518272000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11286\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 20691o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a3, its twist by $33$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/3\Z\) | 2.2.33.1-361.1-d3 |
$3$ | 3.1.76.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.1561115259.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.4683345777.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.2.207572112.3 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | 12.0.168305397188049.1 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.5625641541672730658305241708212224.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.151892321625163727774241526121730048.1 | \(\Z/18\Z\) | 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 | ss | add | ord | ord | add | ord | ord | nonsplit | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2,5 | - | 1 | 1 | - | 3 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-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.