Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-2220x-75056\) | (homogenize, simplify) |
\(y^2z=x^3-2220xz^2-75056z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2220x-75056\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(170, 2112\right)\) | \(\left(368, 6996\right)\) |
$\hat{h}(P)$ | ≈ | $0.55681865798675950485611883444$ | $3.1476529777281051276169037801$ |
Integral points
\((60,\pm 88)\), \((101,\pm 855)\), \((170,\pm 2112)\), \((368,\pm 6996)\), \((1578,\pm 62656)\)
Invariants
Conductor: | \( 145728 \) | = | $2^{6} \cdot 3^{2} \cdot 11 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1733403082752 $ | = | $-1 \cdot 2^{21} \cdot 3^{3} \cdot 11^{3} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{170953875}{244904} \) | = | $-1 \cdot 2^{-3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{-3} \cdot 23^{-1} \cdot 37^{3}$ | 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: | $1.0405369581651782348887055397\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $-0.27383688484176715208595395172\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.849575513642006\dots$ | |||
Szpiro ratio: | $3.0257902959747316\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.6852840475966484033292491627\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.33063728121897943635268907099\dots$ | 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: | $ 24 $ = $ 2^{2}\cdot2\cdot3\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)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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! $ ≈ $ 13.373225653897751127040002693 $ | 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
$\displaystyle 13.373225654 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.330637 \cdot 1.685284 \cdot 24}{1^2} \approx 13.373225654$
Modular invariants
Modular form 145728.2.a.ec
For more coefficients, see the Downloads section to the right.
Modular degree: | 165888 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \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 4 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 6 | 21 | 3 |
$3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
$11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
$23$ | $1$ | $I_{1}$ | 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 | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6072 = 2^{3} \cdot 3 \cdot 11 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 3313 & 6 \\ 3867 & 19 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 4302 & 1777 \\ 4051 & 5838 \end{array}\right),\left(\begin{array}{rr} 3961 & 6 \\ 5811 & 19 \end{array}\right),\left(\begin{array}{rr} 3035 & 6066 \\ 3033 & 6053 \end{array}\right),\left(\begin{array}{rr} 4553 & 6066 \\ 0 & 6071 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6067 & 6 \\ 6066 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6072])$ is a degree-$16250653900800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6072\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | additive | $4$ | \( 759 = 3 \cdot 11 \cdot 23 \) |
$3$ | additive | $6$ | \( 1472 = 2^{6} \cdot 23 \) |
$11$ | split multiplicative | $12$ | \( 13248 = 2^{6} \cdot 3^{2} \cdot 23 \) |
$23$ | split multiplicative | $24$ | \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 145728.ec
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4554.h1, its twist by $24$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.6072.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.223869685248.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.313350280704.6 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.1179813888.1 | \(\Z/6\Z\) | 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 |
$18$ | 18.0.434672743777115126610754837596921605152737067008.2 | \(\Z/9\Z\) | not in database |
$18$ | 18.2.1845363903244761219733862156231098547960807424.1 | \(\Z/6\Z\) | 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 | add | ss | ord | split | ord | ord | ord | split | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | - | - | 4,2 | 2 | 3 | 2 | 2 | 2 | 3 | 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 |
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.