Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-1455x+20725\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-1455xz^2+20725z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1885707x+995239494\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(13, 58\right)\) |
$\hat{h}(P)$ | ≈ | $1.0870549878617145167020845697$ |
Torsion generators
\( \left(23, -2\right) \)
Integral points
\( \left(-37, 178\right) \), \( \left(-37, -142\right) \), \( \left(-7, 178\right) \), \( \left(-7, -172\right) \), \( \left(13, 58\right) \), \( \left(13, -72\right) \), \( \left(19, 10\right) \), \( \left(19, -30\right) \), \( \left(23, -2\right) \), \( \left(23, -22\right) \), \( \left(43, 178\right) \), \( \left(43, -222\right) \)
Invariants
Conductor: | \( 1790 \) | = | $2 \cdot 5 \cdot 179$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $572800000 $ | = | $2^{10} \cdot 5^{5} \cdot 179 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{340668004990321}{572800000} \) | = | $2^{-10} \cdot 5^{-5} \cdot 179^{-1} \cdot 211^{3} \cdot 331^{3}$ | 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: | $0.57577450335243981361135629011\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.57577450335243981361135629011\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9174672531944684\dots$ | |||
Szpiro ratio: | $4.467564691308079\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.0870549878617145167020845697\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $1.6357709838653535161645180198\dots$ | 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: | $ 50 $ = $ ( 2 \cdot 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) $ ≈ $ 3.5563460140205933581229018151 $ | 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
$\displaystyle 3.556346014 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.635771 \cdot 1.087055 \cdot 50}{5^2} \approx 3.556346014$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 1200 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$179$ | $1$ | $I_{1}$ | 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 \( 3580 = 2^{2} \cdot 5 \cdot 179 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1086 & 5 \\ 2675 & 3576 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3525 & 3461 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 1785 & 3576 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3571 & 10 \\ 3570 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 1795 \\ 1785 & 712 \end{array}\right)$.
The torsion field $K:=\Q(E[3580])$ is a degree-$980024140800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3580\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 1790.c
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 |
---|---|---|---|
$3$ | 3.3.3580.1 | \(\Z/10\Z\) | Not in database |
$6$ | 6.6.45882712000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | deg 8 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.33899911210766020745118972320274814483642578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 179 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ord | split | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | 6 | 1 | 2 | 1 | 3 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.