Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-12142535x+16280729097\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-12142535xz^2+16280729097z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15736725387x+759640906925766\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{10}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1984, -317)$ | $1.4765421049243784601663971762$ | $\infty$ |
| $(2284, 20383)$ | $0$ | $10$ |
Integral points
\( \left(-3866, 75733\right) \), \( \left(-3866, -71867\right) \), \( \left(-2336, 179773\right) \), \( \left(-2336, -177437\right) \), \( \left(-866, 162133\right) \), \( \left(-866, -161267\right) \), \( \left(562, 97873\right) \), \( \left(562, -98435\right) \), \( \left(1654, 26053\right) \), \( \left(1654, -27707\right) \), \( \left(1822, 13453\right) \), \( \left(1822, -15275\right) \), \( \left(1984, -317\right) \), \( \left(1984, -1667\right) \), \( \left(2038, -1019\right) \), \( \left(2074, 3373\right) \), \( \left(2074, -5447\right) \), \( \left(2284, 20383\right) \), \( \left(2284, -22667\right) \), \( \left(3334, 111733\right) \), \( \left(3334, -115067\right) \), \( \left(10894, 1079413\right) \), \( \left(10894, -1090307\right) \), \( \left(56794, 13481593\right) \), \( \left(56794, -13538387\right) \)
Invariants
| Conductor: | $N$ | = | \( 94710 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 41$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $58723661173161600000$ | = | $2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 7^{5} \cdot 11 \cdot 41^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( \frac{197993898174778925173824241}{58723661173161600000} \) | = | $2^{-10} \cdot 3^{-10} \cdot 5^{-5} \cdot 7^{-5} \cdot 11^{-1} \cdot 29^{3} \cdot 41^{-2} \cdot 331^{3} \cdot 60719^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7718533712944380346538090522$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7718533712944380346538090522$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $0.9779283874234155$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.2842765466535155$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4765421049243784601663971762$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $0.19347675744064784732588769975$ | 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: | $\prod_{p}c_p$ | = | $ 5000 $ = $ ( 2 \cdot 5 )\cdot( 2 \cdot 5 )\cdot5\cdot5\cdot1\cdot2 $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $10$ | 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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| Special value: | $ L'(E,1)$ | ≈ | $14.283828934267878736730529123 $ | 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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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 14.283828934 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.193477 \cdot 1.476542 \cdot 5000}{10^2} \approx 14.283828934$
Modular invariants
Modular form 94710.2.a.cx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6720000 | 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 at primes of bad reduction
This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $41$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
| $5$ | 5B.1.1 | 5.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 378840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 41 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 216486 & 13 \\ 216395 & 378656 \end{array}\right),\left(\begin{array}{rr} 75786 & 13 \\ 155 & 112 \end{array}\right),\left(\begin{array}{rr} 94711 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 126281 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 378821 & 20 \\ 378820 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 378600 & 378491 \end{array}\right),\left(\begin{array}{rr} 172206 & 13 \\ 275435 & 378656 \end{array}\right),\left(\begin{array}{rr} 189421 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 129361 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[378840])$ is a degree-$9009460622131200000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/378840\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 385 = 5 \cdot 7 \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 31570 = 2 \cdot 5 \cdot 7 \cdot 11 \cdot 41 \) |
| $5$ | split multiplicative | $6$ | \( 451 = 11 \cdot 41 \) |
| $7$ | split multiplicative | $8$ | \( 13530 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 41 \) |
| $11$ | split multiplicative | $12$ | \( 8610 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 94710cy
consists of 4 curves linked by isogenies of
degrees dividing 10.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{385}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $4$ | 4.4.372778560.3 | \(\Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/30\Z\) | not in database |
| $16$ | deg 16 | \(\Z/40\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/30\Z\) | not in database |
| $20$ | 20.0.89407661343122988478808492367662580240087921142578125.1 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | split | split | split | ord | ord | ss | ord | ss | ord | ord | split | ord | ord |
| $\lambda$-invariant(s) | 3 | 2 | 8 | 2 | 2 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 2 | 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.