Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-13736615x+11732181225\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-13736615xz^2+11732181225z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-17802653067x+547430055192774\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{10}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(346, 83617\right)\) |
$\hat{h}(P)$ | ≈ | $2.9530842098487569203327943523$ |
Torsion generators
\( \left(640, 56275\right) \)
Integral points
\( \left(-3920, 75085\right) \), \( \left(-3920, -71165\right) \), \( \left(-1670, 174085\right) \), \( \left(-1670, -172415\right) \), \( \left(346, 83617\right) \), \( \left(346, -83963\right) \), \( \left(640, 56275\right) \), \( \left(640, -56915\right) \), \( \left(3580, 90085\right) \), \( \left(3580, -93665\right) \), \( \left(7570, 580645\right) \), \( \left(7570, -588215\right) \), \( \left(10930, 1074985\right) \), \( \left(10930, -1085915\right) \), \( \left(17580, 2272335\right) \), \( \left(17580, -2289915\right) \), \( \left(227020, 108039535\right) \), \( \left(227020, -108266555\right) \)
Invariants
Conductor: | \( 94710 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 41$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $106415753000070937500000 $ | = | $2^{5} \cdot 3^{5} \cdot 5^{10} \cdot 7^{10} \cdot 11^{2} \cdot 41 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{286657225480543563498731761}{106415753000070937500000} \) | = | $2^{-5} \cdot 3^{-5} \cdot 5^{-10} \cdot 7^{-10} \cdot 11^{-2} \cdot 41^{-1} \cdot 659357521^{3}$ | 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: | $3.1184269615744106893624251130\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.1184269615744106893624251130\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9871615616306024\dots$ | |||
Szpiro ratio: | $5.316571221771278\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.9530842098487569203327943523\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.096738378720323923662943849877\dots$ | 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: | $ 5000 $ = $ 5\cdot5\cdot( 2 \cdot 5 )\cdot( 2 \cdot 5 )\cdot2\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: | $10$ | 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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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) $ ≈ $ 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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BSD formula
$\displaystyle 14.283828934 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.096738 \cdot 2.953084 \cdot 5000}{10^2} \approx 14.283828934$
Modular invariants
Modular form 94710.2.a.cx
For more coefficients, see the Downloads section to the right.
Modular degree: | 13440000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$3$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$5$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$7$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$41$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$2$ | 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
$\left(\begin{array}{rr} 175576 & 5 \\ 240195 & 378826 \end{array}\right),\left(\begin{array}{rr} 378821 & 20 \\ 378820 & 21 \end{array}\right),\left(\begin{array}{rr} 16 & 5 \\ 189375 & 378826 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 162361 & 20 \\ 108250 & 201 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 378600 & 378491 \end{array}\right),\left(\begin{array}{rr} 344401 & 20 \\ 34450 & 201 \end{array}\right),\left(\begin{array}{rr} 189426 & 5 \\ 284185 & 46 \end{array}\right),\left(\begin{array}{rr} 75771 & 20 \\ 190 & 1267 \end{array}\right),\left(\begin{array}{rr} 126296 & 5 \\ 378795 & 378826 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right)$.
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 $\GL_2(\Z/378840\Z)$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{246}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$4$ | 4.0.583413600.8 | \(\Z/20\Z\) | 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) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.