Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-75969x-8031807\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-75969xz^2-8031807z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6153516x-5873647824\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-159, 48\right)\) | \(\left(-157, 28\right)\) |
$\hat{h}(P)$ | ≈ | $1.3710896996637713017346652332$ | $2.1583509521394841816238938139$ |
Torsion generators
\( \left(-161, 0\right) \)
Integral points
\( \left(-161, 0\right) \), \((-159,\pm 48)\), \((-157,\pm 28)\), \((323,\pm 1012)\), \((368,\pm 3703)\), \((417,\pm 5712)\), \((897,\pm 25392)\), \((1761,\pm 72912)\)
Invariants
Conductor: | \( 30912 \) | = | $2^{6} \cdot 3 \cdot 7 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $16175579922432 $ | = | $2^{17} \cdot 3^{2} \cdot 7^{2} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{369937818893666}{123409881} \) | = | $2 \cdot 3^{-2} \cdot 7^{-2} \cdot 23^{-4} \cdot 227^{3} \cdot 251^{3}$ | 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: | $1.5072434134032473625548551923\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.52528490760999150754710968690\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.8686571822608831197110353278\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.28757337238217703494856449265\dots$ | 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: | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ | 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: | $2$ | 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! $ ≈ $ 6.5995953608889251090684013154 $ | 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
$\displaystyle 6.599595361 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.287573 \cdot 2.868657 \cdot 32}{2^2} \approx 6.599595361$
Modular invariants
Modular form 30912.2.a.k
For more coefficients, see the Downloads section to the right.
Modular degree: | 131072 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{7}^{*}$ | Additive | 1 | 6 | 17 | 0 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$23$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 | 8.24.0.58 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 184 = 2^{3} \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 177 & 8 \\ 176 & 9 \end{array}\right),\left(\begin{array}{rr} 123 & 116 \\ 162 & 29 \end{array}\right),\left(\begin{array}{rr} 97 & 8 \\ 20 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 178 & 179 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 112 & 15 \\ 77 & 90 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[184])$ is a degree-$8549376$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/184\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 30912j
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3864e3, its twist by $8$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | Not in database |
$4$ | 4.2.903168.3 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.3262849744896.37 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.11008380780544.18 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.380291185115136.70 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | ord | split | ss | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | - | 4 | 2 | 3 | 2,2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 | 4,2 |
$\mu$-invariant(s) | - | 0 | 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.