Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-20320943x+8757439031\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-20320943xz^2+8757439031z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-325135083x+560150962918\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(403, 24966\right)\) |
$\hat{h}(P)$ | ≈ | $3.1842751274145747421052079683$ |
Torsion generators
\( \left(435, -218\right) \), \( \left(-477, 135670\right) \)
Integral points
\( \left(-4635, 60232\right) \), \( \left(-4635, -55598\right) \), \( \left(-2985, 208402\right) \), \( \left(-2985, -205418\right) \), \( \left(-2453, 210606\right) \), \( \left(-2453, -208154\right) \), \( \left(-477, 135670\right) \), \( \left(-477, -135194\right) \), \( \left(403, 24966\right) \), \( \left(403, -25370\right) \), \( \left(435, -218\right) \), \( \left(4275, -2138\right) \), \( \left(6211, 346342\right) \), \( \left(6211, -352554\right) \), \( \left(7275, 492262\right) \), \( \left(7275, -499538\right) \), \( \left(22095, 3205462\right) \), \( \left(22095, -3227558\right) \), \( \left(33381, 6026962\right) \), \( \left(33381, -6060344\right) \), \( \left(617745, 485205442\right) \), \( \left(617745, -485823188\right) \)
Invariants
Conductor: | \( 18810 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $503953215403595858841600 $ | = | $2^{12} \cdot 3^{10} \cdot 5^{2} \cdot 11^{6} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1272998045160051207059881}{691293848290254950400} \) | = | $2^{-12} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{3} \cdot 11^{-6} \cdot 13^{3} \cdot 19^{-6} \cdot 43^{3} \cdot 27697^{3}$ | 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: | $3.2389773596166752997831025838\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.6896712152826204540854799653\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.035453641620829\dots$ | |||
Szpiro ratio: | $6.309102085691038\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.1842751274145747421052079683\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.081067475383686706554021322396\dots$ | 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: | $ 3456 $ = $ ( 2^{2} \cdot 3 )\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 ) $ | 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: | $12$ | 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) $ ≈ $ 6.1953874921576053345799245718 $ | 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
$\displaystyle 6.195387492 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.081067 \cdot 3.184275 \cdot 3456}{12^2} \approx 6.195387492$
Modular invariants
Modular form 18810.2.a.p
For more coefficients, see the Downloads section to the right.
Modular degree: | 3096576 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$19$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
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 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12540 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9121 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6277 & 6 \\ 12492 & 12499 \end{array}\right),\left(\begin{array}{rr} 9901 & 12 \\ 9246 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 12529 & 12 \\ 12528 & 13 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 12524 & 12533 \end{array}\right),\left(\begin{array}{rr} 10039 & 6 \\ 7518 & 12535 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8359 & 12528 \\ 8360 & 12539 \end{array}\right)$.
The torsion field $K:=\Q(E[12540])$ is a degree-$9361059840000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12540\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 18810.p
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 6270.l3, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{15}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-33}, \sqrt{-57})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.110716875.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.47836184118388200663480000.2 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 | add | nonsplit | ord | split | ord | ord | split | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 5 | - | 1 | 1 | 2 | 1 | 1 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 1 | - | 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.