Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-2040x+69696\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-2040xz^2+69696z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2643867x+3259668150\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(0, 264\right)\) |
$\hat{h}(P)$ | ≈ | $1.1326667411675158287049090619$ |
Torsion generators
\( \left(48, 264\right) \)
Integral points
\( \left(-48, 264\right) \), \( \left(-48, -216\right) \), \( \left(-24, 336\right) \), \( \left(-24, -312\right) \), \( \left(0, 264\right) \), \( \left(0, -264\right) \), \( \left(30, 174\right) \), \( \left(30, -204\right) \), \( \left(48, 264\right) \), \( \left(48, -312\right) \), \( \left(84, 660\right) \), \( \left(84, -744\right) \), \( \left(158, 1844\right) \), \( \left(158, -2002\right) \), \( \left(624, 15240\right) \), \( \left(624, -15864\right) \)
Invariants
Conductor: | \( 4854 \) | = | $2 \cdot 3 \cdot 809$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1565348364288 $ | = | $-1 \cdot 2^{15} \cdot 3^{10} \cdot 809 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{938917686360961}{1565348364288} \) | = | $-1 \cdot 2^{-15} \cdot 3^{-10} \cdot 181^{3} \cdot 541^{3} \cdot 809^{-1}$ | 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: | $1.0307366882104779127141290369\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.0307366882104779127141290369\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9874846966021175\dots$ | |||
Szpiro ratio: | $4.221686937682283\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.1326667411675158287049090619\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.75762475436763891658448951147\dots$ | 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: | $ 150 $ = $ ( 3 \cdot 5 )\cdot( 2 \cdot 5 )\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) $ ≈ $ 5.1488181693445993565266367428 $ | 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
$\displaystyle 5.148818169 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.757625 \cdot 1.132667 \cdot 150}{5^2} \approx 5.148818169$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 14400 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$3$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$809$ | $1$ | $I_{1}$ | 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 \( 32360 = 2^{3} \cdot 5 \cdot 809 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 24271 & 16190 \\ 0 & 11327 \end{array}\right),\left(\begin{array}{rr} 32351 & 10 \\ 32350 & 11 \end{array}\right),\left(\begin{array}{rr} 9726 & 5 \\ 8075 & 32356 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 16175 & 32356 \end{array}\right),\left(\begin{array}{rr} 16186 & 5 \\ 8085 & 32356 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 32305 & 32241 \end{array}\right)$.
The torsion field $K:=\Q(E[32360])$ is a degree-$6571242248601600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 4854.b
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 |
---|---|---|---|
$3$ | 3.1.6472.1 | \(\Z/10\Z\) | Not in database |
$6$ | 6.0.271091266048.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | 8.2.75880098917663067.3 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.1027368890527501360730273140364025718227696503936767578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 809 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | 9 | 6 | 3 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.