Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-1692x-27656\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-1692xz^2-27656z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-27075x-1797058\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 58482 \) | = | $2 \cdot 3^{4} \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-30485730888 $ | = | $-1 \cdot 2^{3} \cdot 3^{4} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{140625}{8} \) | = | $-1 \cdot 2^{-3} \cdot 3^{2} \cdot 5^{6}$ | 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: | $0.76863249393320375928267759494\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0697910918727197011869178666\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1780968639246816\dots$ | |||
Szpiro ratio: | $3.0983265977624996\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.37095882917236617216817231836\dots$ | 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: | $ 1 $ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.37095882917236617216817231836 $ | 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
$\displaystyle 0.370958829 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.370959 \cdot 1.000000 \cdot 1}{1^2} \approx 0.370958829$
Modular invariants
Modular form 58482.2.a.g
For more coefficients, see the Downloads section to the right.
Modular degree: | 43092 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $1$ | $II$ | Additive | 1 | 4 | 4 | 0 |
$19$ | $1$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 3192 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 3306 \\ 7980 & 799 \end{array}\right),\left(\begin{array}{rr} 1597 & 5586 \\ 1197 & 5587 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 7980 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1938 \\ 5586 & 4789 \end{array}\right),\left(\begin{array}{rr} 799 & 5586 \\ 6783 & 8779 \end{array}\right),\left(\begin{array}{rr} 4713 & 9082 \\ 8512 & 2433 \end{array}\right),\left(\begin{array}{rr} 1 & 2736 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7582 & 2337 \\ 1197 & 3193 \end{array}\right),\left(\begin{array}{rr} 8567 & 0 \\ 0 & 9575 \end{array}\right),\left(\begin{array}{rr} 6385 & 3192 \\ 6384 & 3193 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3192 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 5586 \\ 0 & 4105 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1008 & 1 \end{array}\right),\left(\begin{array}{rr} 8569 & 1008 \\ 8568 & 8569 \end{array}\right),\left(\begin{array}{rr} 5321 & 7448 \\ 2128 & 7449 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3990 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9576])$ is a degree-$1930080337920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9576\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 58482.g
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162.b3, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-19}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.648.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.1215051273.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.45001899.1 | \(\Z/21\Z\) | Not in database |
$6$ | 6.0.2880121536.12 | \(\Z/6\Z\) | 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 |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.1785459633425225745882684908875776.7 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.342808249617643343209475502504148992.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.0.23890896332218429125610438656.1 | \(\Z/42\Z\) | 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 | ss | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | - | 0,0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.