Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-187880x-31262199\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-187880xz^2-31262199z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-243492507x-1454916760650\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{5}\Z\)
Torsion generators
\( \left(579, 7091\right) \)
Integral points
\( \left(579, 7091\right) \), \( \left(579, -7671\right) \), \( \left(1921, 80901\right) \), \( \left(1921, -82823\right) \)
Invariants
Conductor: | \( 1342 \) | = | $2 \cdot 11 \cdot 61$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $4352738523915232 $ | = | $2^{5} \cdot 11^{5} \cdot 61^{5} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{733441552889589371521}{4352738523915232} \) | = | $2^{-5} \cdot 11^{-5} \cdot 31^{3} \cdot 61^{-5} \cdot 421^{3} \cdot 691^{3}$ | 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: | $1.8406811932579279750715705706\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.8406811932579279750715705706\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.028282599584136\dots$ | |||
Szpiro ratio: | $6.671041076782771\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.22939536195784025899090284570\dots$ | 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: | $ 125 $ = $ 5\cdot5\cdot5 $ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.1469768097892012949545142285 $ | 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
$\displaystyle 1.146976810 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.229395 \cdot 1.000000 \cdot 125}{5^2} \approx 1.146976810$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 16000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$11$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$61$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
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 |
---|---|---|
$5$ | 5Cs.1.1 | 5.120.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 134200 = 2^{3} \cdot 5^{2} \cdot 11 \cdot 61 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 16 & 35 \\ 1765 & 3861 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 50631 & 50 \\ 14335 & 23121 \end{array}\right),\left(\begin{array}{rr} 134151 & 50 \\ 134150 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 67101 & 50 \\ 67125 & 1251 \end{array}\right),\left(\begin{array}{rr} 122021 & 50 \\ 84615 & 132331 \end{array}\right),\left(\begin{array}{rr} 18801 & 50 \\ 54104 & 89377 \end{array}\right),\left(\begin{array}{rr} 33551 & 50 \\ 33575 & 1251 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right)$.
The torsion field $K:=\Q(E[134200])$ is a degree-$69012725760000000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/134200\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 1342c
consists of 3 curves linked by isogenies of
degrees dividing 25.
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 |
---|---|---|---|
$3$ | 3.3.5368.1 | \(\Z/10\Z\) | Not in database |
$4$ | \(\Q(\zeta_{5})\) | \(\Z/5\Z \oplus \Z/5\Z\) | Not in database |
$6$ | 6.6.154681196032.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | deg 8 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$12$ | deg 12 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 | 11 | 61 |
---|---|---|---|---|---|
Reduction type | split | ord | ord | split | split |
$\lambda$-invariant(s) | 10 | 0 | 2 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.