Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+292714x+54762104\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+292714xz^2+54762104z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+379357965x+2553842661966\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-63, 6037\right)\)
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| $\hat{h}(P)$ | ≈ | $0.93211303552130654193357309234$ |
Torsion generators
\( \left(1224, 46792\right) \)
Integral points
\( \left(-63, 6037\right) \), \( \left(-63, -5975\right) \), \( \left(102, 9205\right) \), \( \left(102, -9308\right) \), \( \left(234, 11548\right) \), \( \left(234, -11783\right) \), \( \left(561, 19609\right) \), \( \left(561, -20171\right) \), \( \left(1224, 46792\right) \), \( \left(1224, -48017\right) \), \( \left(3369, 196513\right) \), \( \left(3369, -199883\right) \), \( \left(8517, 783385\right) \), \( \left(8517, -791903\right) \), \( \left(24786, 3890764\right) \), \( \left(24786, -3915551\right) \)
Invariants
| Conductor: | \( 14586 \) | = | $2 \cdot 3 \cdot 11 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-2899504554614368272 $ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 11^{6} \cdot 13^{4} \cdot 17^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{2773679829880629422375}{2899504554614368272} \) | = | $2^{-4} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-6} \cdot 13^{-4} \cdot 17^{-3} \cdot 229^{3} \cdot 1753^{3}$ | 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: | $2.2293332654650940733062542130\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.2293332654650940733062542130\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.93211303552130654193357309234\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.16808338249122457901756708971\dots$ | 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: | $ 864 $ = $ 2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot3 $ | 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: | $6$ | 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) $ ≈ $ 3.7601450849900200823438435907 $ | 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
$\displaystyle 3.760145085 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.168083 \cdot 0.932113 \cdot 864}{6^2} \approx 3.760145085$
Modular invariants
Modular form 14586.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 304128 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $11$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2244 = 2^{2} \cdot 3 \cdot 11 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2058 & 1321 \\ 2057 & 1310 \end{array}\right),\left(\begin{array}{rr} 1594 & 3 \\ 633 & 2236 \end{array}\right),\left(\begin{array}{rr} 409 & 12 \\ 210 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2194 & 2235 \end{array}\right),\left(\begin{array}{rr} 1505 & 2 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2233 & 12 \\ 2232 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[2244])$ is a degree-$2872340$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2244\Z)$.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | ss | ord | split | split | split | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 7 | 4 | 1,1 | 1 | 2 | 2 | 2 | 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 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 14586.e
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $4$ | 4.2.74052.4 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.12338352.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $8$ | 8.0.25356622807296.4 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $9$ | 9.3.280389568253455762358173488.1 | \(\Z/18\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.0.6328348786001025039997966997346099142744003920746925678133248.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.