Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-1006502x-375717234\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1006502xz^2-375717234z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1304427267x-17509896863874\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-593, 3909\right)\) |
$\hat{h}(P)$ | ≈ | $1.5523077831257202003026870055$ |
Torsion generators
\( \left(-\frac{2661}{4}, \frac{2661}{8}\right) \)
Integral points
\( \left(-593, 3909\right) \), \( \left(-593, -3316\right) \), \( \left(1227, 14829\right) \), \( \left(1227, -16056\right) \), \( \left(3757, 219234\right) \), \( \left(3757, -222991\right) \)
Invariants
Conductor: | \( 67830 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $4410075563770668750 $ | = | $2 \cdot 3^{3} \cdot 5^{5} \cdot 7^{4} \cdot 17^{4} \cdot 19^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{112763439169361881747561}{4410075563770668750} \) | = | $2^{-1} \cdot 3^{-3} \cdot 5^{-5} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{3} \cdot 17^{-4} \cdot 19^{-4} \cdot 23^{3} \cdot 37^{3} \cdot 397^{3}$ | 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: | $2.3447302995939071838923532437\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.3447302995939071838923532437\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9594529041261464\dots$ | |||
Szpiro ratio: | $4.771301117098891\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5523077831257202003026870055\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.15109188547287575190860139374\dots$ | 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: | $ 80 $ = $ 1\cdot1\cdot5\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'(E,1) $ ≈ $ 4.6908221957336993437091926526 $ | 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
$\displaystyle 4.690822196 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.151092 \cdot 1.552308 \cdot 80}{2^2} \approx 4.690822196$
Modular invariants
Modular form 67830.2.a.k
For more coefficients, see the Downloads section to the right.
Modular degree: | 1597440 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$17$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$19$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 271320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 237408 & 101753 \\ 33907 & 33894 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 169576 & 33923 \\ 169579 & 169608 \end{array}\right),\left(\begin{array}{rr} 79801 & 8 \\ 47884 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 271314 & 271315 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 99961 & 8 \\ 128524 & 33 \end{array}\right),\left(\begin{array}{rr} 271313 & 8 \\ 271312 & 9 \end{array}\right),\left(\begin{array}{rr} 108536 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 77521 & 8 \\ 38764 & 33 \end{array}\right),\left(\begin{array}{rr} 180884 & 1 \\ 90463 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[271320])$ is a degree-$14335504436271513600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/271320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 67830j
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{-6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.47775744000000.32 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | split | ss | ord | nonsplit | nonsplit | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 7 | 1 | 6 | 2 | 1,1 | 1 | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 2 | 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.