Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-43929x-11586515\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-43929xz^2-11586515z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-702867x-742239826\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(461, 7892\right)\) | \(\left(866, 24047\right)\) |
$\hat{h}(P)$ | ≈ | $1.3202412374542542366736045174$ | $2.6823932503526549916550286740$ |
Torsion generators
\( \left(290, -145\right) \)
Integral points
\( \left(290, -145\right) \), \( \left(411, 6092\right) \), \( \left(411, -6503\right) \), \( \left(461, 7892\right) \), \( \left(461, -8353\right) \), \( \left(651, 15017\right) \), \( \left(651, -15668\right) \), \( \left(866, 24047\right) \), \( \left(866, -24913\right) \), \( \left(1506, 57007\right) \), \( \left(1506, -58513\right) \), \( \left(5154, 367087\right) \), \( \left(5154, -372241\right) \), \( \left(14141, 1674287\right) \), \( \left(14141, -1688428\right) \)
Invariants
Conductor: | \( 32490 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-52679342974464000 $ | = | $-1 \cdot 2^{12} \cdot 3^{7} \cdot 5^{3} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{273359449}{1536000} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-1} \cdot 5^{-3} \cdot 11^{3} \cdot 59^{3}$ | 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: | $1.8925444385736814465741465345\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.12898119534359362912798979991\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0491971880149842\dots$ | |||
Szpiro ratio: | $4.433259228326581\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.4909759124137503597761230635\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14799213095244032661588969214\dots$ | 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: | $ 48 $ = $ 2\cdot2\cdot3\cdot2^{2} $ | 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: | $2$ | 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! $ ≈ $ 6.1996435725810071431090883378 $ | 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
$\displaystyle 6.199643573 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.147992 \cdot 3.490976 \cdot 48}{2^2} \approx 6.199643573$
Modular invariants
Modular form 32490.2.a.n
For more coefficients, see the Downloads section to the right.
Modular degree: | 331776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$3$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$19$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
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 | 4.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2257 & 24 \\ 2256 & 25 \end{array}\right),\left(\begin{array}{rr} 1312 & 1083 \\ 1197 & 514 \end{array}\right),\left(\begin{array}{rr} 457 & 1824 \\ 114 & 1711 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 974 & 2171 \end{array}\right),\left(\begin{array}{rr} 2184 & 247 \\ 1729 & 246 \end{array}\right),\left(\begin{array}{rr} 609 & 988 \\ 380 & 989 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 359 & 0 \\ 0 & 2279 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$11346739200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 32490ba
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a3, its twist by $57$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{285}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-19}) \) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.2.1215051273.1 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.380016036000000.43 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.675584064000000.44 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.9728410521600.12 | \(\Z/24\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | 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 |
$18$ | 18.0.66438535408734263735084571000000000000.5 | \(\Z/36\Z\) | 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 | nonsplit | add | split | ord | ss | ord | ord | add | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 3 | - | 3 | 2 | 2,2 | 4 | 2 | - | 2,2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-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.