Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-1134x+19840\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-1134xz^2+19840z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-18147x+1251614\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(6, 112\right)\) | \(\left(14, 74\right)\) |
$\hat{h}(P)$ | ≈ | $0.89280909287554904639513753746$ | $0.96910265720686420183144891728$ |
Torsion generators
\( \left(-40, 20\right) \)
Integral points
\( \left(-40, 20\right) \), \( \left(-34, 152\right) \), \( \left(-34, -118\right) \), \( \left(-15, 190\right) \), \( \left(-15, -175\right) \), \( \left(6, 112\right) \), \( \left(6, -118\right) \), \( \left(9, 97\right) \), \( \left(9, -106\right) \), \( \left(14, 74\right) \), \( \left(14, -88\right) \), \( \left(29, 89\right) \), \( \left(29, -118\right) \), \( \left(41, 182\right) \), \( \left(41, -223\right) \), \( \left(56, 332\right) \), \( \left(56, -388\right) \), \( \left(581, 13682\right) \), \( \left(581, -14263\right) \), \( \left(1310, 46730\right) \), \( \left(1310, -48040\right) \)
Invariants
Conductor: | \( 47610 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 23^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-71844918300 $ | = | $-1 \cdot 2^{2} \cdot 3^{10} \cdot 5^{2} \cdot 23^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{18191447}{8100} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-4} \cdot 5^{-2} \cdot 263^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.79007872307712039918152338036\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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||
Stable Faltings height: | $-0.54310097523922186921778744605\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0401575600846762\dots$ | |||
Szpiro ratio: | $3.090325155841391\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.86458583946223704832890945999\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.0229242570128343486889680628\dots$ | 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: | $ 32 $ = $ 2\cdot2^{2}\cdot2\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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 7.0752466196458120668394025201 $ | 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
$\displaystyle 7.075246620 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.022924 \cdot 0.864586 \cdot 32}{2^2} \approx 7.075246620$
Modular invariants
Modular form 47610.2.a.v
For more coefficients, see the Downloads section to the right.
Modular degree: | 61440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$23$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
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 | 8.6.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 184 = 2^{3} \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 73 & 116 \\ 160 & 23 \end{array}\right),\left(\begin{array}{rr} 93 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 181 & 4 \\ 180 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20 & 1 \\ 87 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[184])$ is a degree-$34197504$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/184\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 47610.v
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 15870.x2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.7008192.3 | \(\Z/4\Z\) | Not in database |
$8$ | 8.0.7674180485760000.25 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.49114755108864.48 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | split | ss | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | 3 | 2,2 | 2 | 2 | 2 | 4 | - | 2 | 2 | 2 | 2 | 4 | 2 |
$\mu$-invariant(s) | 0 | - | 0 | 0,0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.