Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-72x+496\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-72xz^2+496z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-5859x+344034\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(36, 208\right)\) |
$\hat{h}(P)$ | ≈ | $0.055429397200046597798380261449$ |
Integral points
\((-6,\pm 26)\), \((-3,\pm 26)\), \((4,\pm 16)\), \((10,\pm 26)\), \((36,\pm 208)\)
Invariants
Conductor: | \( 208 \) | = | $2^{4} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-71991296 $ | = | $-1 \cdot 2^{15} \cdot 13^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{10218313}{17576} \) | = | $-1 \cdot 2^{-3} \cdot 7^{3} \cdot 13^{-3} \cdot 31^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.19822746235926711949282001369\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.49491971820067818992441210777\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.947172997962475\dots$ | |||
Szpiro ratio: | $4.840581786050342\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.055429397200046597798380261449\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.7396717417157566288177692679\dots$ | 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: | $ 12 $ = $ 2^{2}\cdot3 $ | 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) $ ≈ $ 1.1571474716311145826040968992 $ | 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
$\displaystyle 1.157147472 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.739672 \cdot 0.055429 \cdot 12}{1^2} \approx 1.157147472$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 48 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{7}^{*}$ | Additive | -1 | 4 | 15 | 3 |
$13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 |
---|---|---|
$3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 580 & 9 \\ 567 & 916 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 713 & 486 \\ 306 & 541 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 918 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 458 & 927 \\ 711 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 459 & 928 \end{array}\right)$.
The torsion field $K:=\Q(E[936])$ is a degree-$1086898176$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/936\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 208.a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26.a2, its twist by $-4$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | 2.2.12.1-338.1-d2 |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/3\Z\) | 2.0.4.1-338.2-a2 |
$3$ | 3.1.104.1 | \(\Z/2\Z\) | Not in database |
$4$ | \(\Q(\zeta_{12})\) | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.4672512.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.173056.2 | \(\Z/6\Z\) | Not in database |
$12$ | 12.2.8421963387109376.3 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | 12.0.80980417183744.8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.6.2980670717600796756944181214052352.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.2366152515496237836064260096.1 | \(\Z/9\Z\) | 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 | add | ord | ord | ord | ord | split | ord | ord | ss | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | - | 1 | 3 | 1 | 1 | 2 | 1 | 1 | 1,1 | 1 | 1 | 3 | 1,1 | 1 | 1 |
$\mu$-invariant(s) | - | 0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.