Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-13442x+503844\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-13442xz^2+503844z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-17421507x+23768664894\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(8, 626\right)\) | \(\left(43, 66\right)\) |
$\hat{h}(P)$ | ≈ | $0.62451263992709976574813330604$ | $1.3533652382525945844837762691$ |
Torsion generators
\( \left(-132, 66\right) \), \( \left(44, -22\right) \)
Integral points
\( \left(-132, 66\right) \), \( \left(-127, 491\right) \), \( \left(-127, -364\right) \), \( \left(-125, 563\right) \), \( \left(-125, -438\right) \), \( \left(-55, 1067\right) \), \( \left(-55, -1012\right) \), \( \left(-32, 966\right) \), \( \left(-32, -934\right) \), \( \left(8, 626\right) \), \( \left(8, -634\right) \), \( \left(43, 66\right) \), \( \left(43, -109\right) \), \( \left(44, -22\right) \), \( \left(88, 66\right) \), \( \left(88, -154\right) \), \( \left(93, 216\right) \), \( \left(93, -309\right) \), \( \left(120, 738\right) \), \( \left(120, -858\right) \), \( \left(143, 1166\right) \), \( \left(143, -1309\right) \), \( \left(188, 2066\right) \), \( \left(188, -2254\right) \), \( \left(253, 3531\right) \), \( \left(253, -3784\right) \), \( \left(368, 6566\right) \), \( \left(368, -6934\right) \), \( \left(748, 19866\right) \), \( \left(748, -20614\right) \), \( \left(1583, 62051\right) \), \( \left(1583, -63634\right) \), \( \left(1893, 81291\right) \), \( \left(1893, -83184\right) \), \( \left(5368, 390566\right) \), \( \left(5368, -395934\right) \), \( \left(7568, 654566\right) \), \( \left(7568, -662134\right) \)
Invariants
Conductor: | \( 43890 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $43342472250000 $ | = | $2^{4} \cdot 3^{4} \cdot 5^{6} \cdot 7^{2} \cdot 11^{2} \cdot 19^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{268637023475732521}{43342472250000} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-2} \cdot 19^{-2} \cdot 607^{3} \cdot 1063^{3}$ | 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: | $1.3388274470979956885082043913\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $1.3388274470979956885082043913\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9127273248509665\dots$ | |||
Szpiro ratio: | $3.7543716804016367\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.83334389634750322777139967265\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.61330331774303685799187600726\dots$ | 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: | $ 192 $ = $ 2\cdot2\cdot( 2 \cdot 3 )\cdot2\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)
|
Torsion order: | $4$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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! $ ≈ $ 6.1331109174099977355499479806 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 6.133110917 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.613303 \cdot 0.833344 \cdot 192}{4^2} \approx 6.133110917$
Modular invariants
Modular form 43890.2.a.l
For more coefficients, see the Downloads section to the right.
Modular degree: | 196608 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$19$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 29260 = 2^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 14633 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 23411 & 2 \\ 23406 & 29259 \end{array}\right),\left(\begin{array}{rr} 1541 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 29257 & 4 \\ 29256 & 5 \end{array}\right),\left(\begin{array}{rr} 25083 & 2 \\ 12538 & 29259 \end{array}\right),\left(\begin{array}{rr} 13301 & 4 \\ 26602 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[29260])$ is a degree-$3145316106240000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29260\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 43890.l
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{11}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-11}, \sqrt{-35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{19}, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | split | nonsplit | split | ord | ord | split | ord | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 2 | 3 | 2 | 3 | 2 | 2 | 3 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 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.