Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-392403x+79782802\)
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(homogenize, simplify) |
\(y^2z=x^3-392403xz^2+79782802z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-392403x+79782802\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = |
\(\left(671, 10890\right)\)
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\(\left(231, 1210\right)\)
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$\hat{h}(P)$ | ≈ | $1.0255379500676956475106935977$ | $2.2732219043046493968897440674$ |
Torsion generators
\( \left(473, 0\right) \)
Integral points
\((-679,\pm 5760)\), \((-418,\pm 13068)\), \((-249,\pm 12730)\), \((231,\pm 1210)\), \( \left(473, 0\right) \), \((594,\pm 7502)\), \((671,\pm 10890)\), \((1881,\pm 77440)\), \((2777,\pm 142848)\), \((8754,\pm 816998)\)
Invariants
Conductor: | \( 87120 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1117216911576268800 $ | = | $2^{20} \cdot 3^{7} \cdot 5^{2} \cdot 11^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1263214441}{211200} \) | = | $2^{-8} \cdot 3^{-1} \cdot 5^{-2} \cdot 11^{-1} \cdot 23^{3} \cdot 47^{3}$ | 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: | $2.1842471768483338858952098227\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.25715378444485154125061670620\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9075122139890862\dots$ | |||
Szpiro ratio: | $4.41789472595383\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.2429496209093765880656962969\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.26270716079928370851075815354\dots$ | 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: | $ 64 $ = $ 2^{2}\cdot2\cdot2\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! $ ≈ $ 9.4278228275989125091915940116 $ | 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
$\displaystyle 9.427822828 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.262707 \cdot 2.242950 \cdot 64}{2^2} \approx 9.427822828$
Modular invariants
Modular form 87120.2.a.d
For more coefficients, see the Downloads section to the right.
Modular degree: | 1474560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{12}^{*}$ | Additive | -1 | 4 | 20 | 8 |
$3$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 |
---|---|---|
$2$ | 2B | 8.12.0.14 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 80 & 261 \\ 83 & 262 \end{array}\right),\left(\begin{array}{rr} 235 & 234 \\ 178 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 258 & 259 \end{array}\right),\left(\begin{array}{rr} 257 & 8 \\ 256 & 9 \end{array}\right),\left(\begin{array}{rr} 95 & 96 \\ 218 & 89 \end{array}\right),\left(\begin{array}{rr} 212 & 263 \\ 97 & 258 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$20275200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 87120.d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 330.b3, its twist by $-132$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{11}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.12914679690000.20 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.8.1707726240000.2 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.36735088896.2 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | 16.0.1349466756197022498816.1 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | nonsplit | ord | add | ord | ord | ord | ss | ord | ord | ord | ord | ss | ss |
$\lambda$-invariant(s) | - | - | 2 | 2 | - | 2 | 2 | 2 | 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 |
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.