Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-254687x+26034999\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-254687xz^2+26034999z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4074987x+1662164966\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-43, 6096\right)\) |
$\hat{h}(P)$ | ≈ | $0.90780346851948134219649687981$ |
Torsion generators
\( \left(107, -54\right) \), \( \left(443, -222\right) \)
Integral points
\( \left(-243, 8696\right) \), \( \left(-243, -8454\right) \), \( \left(-43, 6096\right) \), \( \left(-43, -6054\right) \), \( \left(107, -54\right) \), \( \left(443, -222\right) \), \( \left(507, 4946\right) \), \( \left(507, -5454\right) \), \( \left(737, 15066\right) \), \( \left(737, -15804\right) \), \( \left(1577, 58746\right) \), \( \left(1577, -60324\right) \)
Invariants
Conductor: | \( 4410 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $765912902060250000 $ | = | $2^{4} \cdot 3^{12} \cdot 5^{6} \cdot 7^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{21302308926361}{8930250000} \) | = | $2^{-4} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-2} \cdot 19^{3} \cdot 1459^{3}$ | 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: | $2.1286464905581326785208165160\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.60638527169642118027051752582\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0136164980344768\dots$ | |||
Szpiro ratio: | $5.834024068449731\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.90780346851948134219649687981\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.25676893366714815275551997785\dots$ | 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: | $ 384 $ = $ 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )\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])
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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'(E,1) $ ≈ $ 5.5942974861860573048886765610 $ | 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
$\displaystyle 5.594297486 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.256769 \cdot 0.907803 \cdot 384}{4^2} \approx 5.594297486$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 73728 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$3$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
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 | 4.12.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 409 & 12 \\ 408 & 13 \end{array}\right),\left(\begin{array}{rr} 211 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 413 & 414 \\ 186 & 5 \end{array}\right),\left(\begin{array}{rr} 273 & 410 \\ 278 & 417 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 12 \\ 342 & 73 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 404 & 413 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$11612160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 4410.bi
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210.b6, its twist by $21$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/12\Z\) | 2.2.21.1-2100.1-n5 |
$4$ | \(\Q(i, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$6$ | 6.0.2420208.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.31116960000.8 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.4.8782450790400.27 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$12$ | 12.0.52716660869376.1 | \(\Z/6\Z \oplus \Z/12\Z\) | 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 |
$18$ | 18.6.33663242317883836505645491312500000000.1 | \(\Z/2\Z \oplus \Z/36\Z\) | 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 | split | add | split | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 3 | - | 2 | - | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 1 | - | 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.