Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-280x+1684\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-280xz^2+1684z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-363312x+82938384\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(8, 4\right)\) |
$\hat{h}(P)$ | ≈ | $0.74507484629343581376731957383$ |
Torsion generators
\( \left(11, 7\right) \)
Integral points
\( \left(-19, 22\right) \), \( \left(-19, -23\right) \), \( \left(-4, 52\right) \), \( \left(-4, -53\right) \), \( \left(1, 37\right) \), \( \left(1, -38\right) \), \( \left(8, 4\right) \), \( \left(8, -5\right) \), \( \left(11, 7\right) \), \( \left(11, -8\right) \), \( \left(26, 112\right) \), \( \left(26, -113\right) \), \( \left(47, 307\right) \), \( \left(47, -308\right) \), \( \left(326, 5887\right) \), \( \left(326, -5888\right) \)
Invariants
Conductor: | \( 885 \) | = | $3 \cdot 5 \cdot 59$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $44803125 $ | = | $3^{5} \cdot 5^{5} \cdot 59 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2436396322816}{44803125} \) | = | $2^{12} \cdot 3^{-5} \cdot 5^{-5} \cdot 29^{6} \cdot 59^{-1}$ | 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: | $0.26282189155831665596259892928\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.26282189155831665596259892928\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.058729821515299\dots$ | |||
Szpiro ratio: | $4.203252929408571\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.74507484629343581376731957383\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $2.0244640686484260874014704123\dots$ | 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: | $ 25 $ = $ 5\cdot5\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: | $5$ | 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) $ ≈ $ 1.5083772547748097566395827341 $ | 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
$\displaystyle 1.508377255 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.024464 \cdot 0.745075 \cdot 25}{5^2} \approx 1.508377255$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 400 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$59$ | $1$ | $I_{1}$ | Non-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 |
---|---|---|
$5$ | 5B.1.1 | 5.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1770 = 2 \cdot 3 \cdot 5 \cdot 59 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1761 & 10 \\ 1760 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 5 & 1467 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 10 \\ 305 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1181 & 10 \\ 595 & 51 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 1715 & 1651 \end{array}\right)$.
The torsion field $K:=\Q(E[1770])$ is a degree-$34296652800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1770\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 885.a
consists of 2 curves linked by isogenies of
degree 5.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.3.3540.1 | \(\Z/10\Z\) | Not in database |
$6$ | 6.6.11090466000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | 8.2.1341596343166875.3 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.657933880831189038139583466827392578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 59 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | split | split | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | 4,1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.