Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-431996x+107207286\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-431996xz^2+107207286z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6911931x+6854354390\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(296, 2133\right)\) | \(\left(426, 378\right)\) |
$\hat{h}(P)$ | ≈ | $2.2077699547737504638847538326$ | $2.2504520520578870694186990582$ |
Torsion generators
\( \left(-757, 378\right) \), \( \left(335, -168\right) \)
Integral points
\( \left(-757, 378\right) \), \( \left(-718, 7203\right) \), \( \left(-718, -6486\right) \), \( \left(-394, 14898\right) \), \( \left(-394, -14505\right) \), \( \left(296, 2133\right) \), \( \left(296, -2430\right) \), \( \left(332, 378\right) \), \( \left(332, -711\right) \), \( \left(335, -168\right) \), \( \left(426, 378\right) \), \( \left(426, -805\right) \), \( \left(443, 1398\right) \), \( \left(443, -1842\right) \), \( \left(2792, 142338\right) \), \( \left(2792, -145131\right) \), \( \left(5535, 406212\right) \), \( \left(5535, -411748\right) \), \( \left(8079, 719760\right) \), \( \left(8079, -727840\right) \), \( \left(32276, 5781153\right) \), \( \left(32276, -5813430\right) \)
Invariants
Conductor: | \( 117117 \) | = | $3^{2} \cdot 7 \cdot 11 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $204474653869655169 $ | = | $3^{10} \cdot 7^{2} \cdot 11^{4} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2533811507137}{58110129} \) | = | $3^{-4} \cdot 7^{-2} \cdot 11^{-4} \cdot 13633^{3}$ | 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: | $2.1079972231715761957830487143\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.27621640010675298205868237506\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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||
$abc$ quality: | $0.9546990293922524\dots$ | |||
Szpiro ratio: | $4.330599297142362\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $4.8101058682833073816122233598\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.31654881523554757475208887598\dots$ | 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: | $ 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: | $4$ | 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! $ ≈ $ 6.0905332550505432300201032332 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 6.090533255 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.316549 \cdot 4.810106 \cdot 64}{4^2} \approx 6.090533255$
Modular invariants
Modular form 117117.2.a.l
For more coefficients, see the Downloads section to the right.
Modular degree: | 1228800 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$13$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 4.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 16015 & 3692 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 11087 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18487 & 12480 \\ 5538 & 11545 \end{array}\right),\left(\begin{array}{rr} 4369 & 16640 \\ 13780 & 18513 \end{array}\right),\left(\begin{array}{rr} 18487 & 22178 \\ 2886 & 5539 \end{array}\right),\left(\begin{array}{rr} 24017 & 8 \\ 24016 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 24020 & 24021 \end{array}\right),\left(\begin{array}{rr} 8321 & 16172 \\ 21736 & 4629 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$267811710566400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 117117r
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 231a3, its twist by $-39$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.1421970391296.8 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.2219775733334016.94 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ord | add | ord | nonsplit | nonsplit | add | ord | ord | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | 2 | 2 | 2 | - | 2 | 2 | 2,4 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.