Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-22764x+1320784\) | (homogenize, simplify) |
\(y^2z=x^3-22764xz^2+1320784z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-22764x+1320784\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(26, 864\right)\) | \(\left(80, 108\right)\) |
$\hat{h}(P)$ | ≈ | $0.47025002466053127294803018885$ | $1.1363081543076403257638378959$ |
Integral points
\((-163,\pm 837)\), \((26,\pm 864)\), \((53,\pm 513)\), \((80,\pm 108)\), \((90,\pm 32)\), \((92,\pm 72)\), \((218,\pm 2592)\), \((396,\pm 7376)\), \((2522,\pm 126432)\)
Invariants
Conductor: | \( 75456 \) | = | $2^{6} \cdot 3^{2} \cdot 131$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1351862452224 $ | = | $2^{19} \cdot 3^{9} \cdot 131 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{6826561273}{7074} \) | = | $2^{-1} \cdot 3^{-3} \cdot 7^{3} \cdot 131^{-1} \cdot 271^{3}$ | 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: | $1.2454556970813049026859667178\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.34357121809266790713750408285\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8867469018550661\dots$ | |||
Szpiro ratio: | $3.713941518968469\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.51224026305145848826432579385\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.85242560715567659400828233447\dots$ | 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: | $ 16 $ = $ 2^{2}\cdot2^{2}\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)
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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^{(2)}(E,1)/2! $ ≈ $ 6.9863474758595678969564470503 $ | 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
$\displaystyle 6.986347476 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.852426 \cdot 0.512240 \cdot 16}{1^2} \approx 6.986347476$
Modular invariants
Modular form 75456.2.a.c
For more coefficients, see the Downloads section to the right.
Modular degree: | 233472 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{9}^{*}$ | Additive | -1 | 6 | 19 | 1 |
$3$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$131$ | $1$ | $I_{1}$ | 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 \( 3144 = 2^{3} \cdot 3 \cdot 131 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1705 & 6 \\ 1971 & 19 \end{array}\right),\left(\begin{array}{rr} 1571 & 3138 \\ 1569 & 3125 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3139 & 6 \\ 3138 & 7 \end{array}\right),\left(\begin{array}{rr} 785 & 3138 \\ 0 & 3143 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2226 & 911 \\ 2093 & 2994 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3144])$ is a degree-$1346617958400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3144\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 75456dk
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 786f1, 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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{6}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.3.3144.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.31077609984.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.5428222543872.2 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.948934656.1 | \(\Z/6\Z\) | 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 |
$18$ | 18.6.1035384446109777597587311996666118211108864.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.395255586123327037360716984653004406398124032.1 | \(\Z/6\Z\) | 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 | 131 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | - | - | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |
$\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.