Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+635187x-146561469\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+635187xz^2-146561469z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+823202325x-6850319924250\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(919, 34388\right)\) |
$\hat{h}(P)$ | ≈ | $1.1781591261928335158397279862$ |
Torsion generators
\( \left(215, -108\right) \)
Integral points
\( \left(215, -108\right) \), \( \left(231, 3428\right) \), \( \left(231, -3660\right) \), \( \left(919, 34388\right) \), \( \left(919, -35308\right) \), \( \left(1315, 53792\right) \), \( \left(1315, -55108\right) \), \( \left(48615, 10696292\right) \), \( \left(48615, -10744908\right) \)
Invariants
Conductor: | \( 127050 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-25714562227200000000 $ | = | $-1 \cdot 2^{16} \cdot 3^{4} \cdot 5^{8} \cdot 7 \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1023887723039}{928972800} \) | = | $2^{-16} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-1} \cdot 10079^{3}$ | 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: | $2.4118016483919078874499254601\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.40813505577567242811857400451\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.999810038071446\dots$ | |||
Szpiro ratio: | $4.3990086015202206\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.1781591261928335158397279862\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.11620586345100516920230167634\dots$ | 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: | $ 256 $ = $ 2^{4}\cdot2\cdot2\cdot1\cdot2^{2} $ | 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) $ ≈ $ 8.7621759066828786674761301084 $ | 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
$\displaystyle 8.762175907 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.116206 \cdot 1.178159 \cdot 256}{2^2} \approx 8.762175907$
Modular invariants
Modular form 127050.2.a.fh
For more coefficients, see the Downloads section to the right.
Modular degree: | 3932160 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$5$ | $2$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 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 | 16.48.0.96 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 36960 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 23 & 18 \\ 34398 & 34955 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 7369 & 10054 \\ 31482 & 2485 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 8823 & 13442 \\ 32362 & 20175 \end{array}\right),\left(\begin{array}{rr} 5281 & 30272 \\ 11594 & 34079 \end{array}\right),\left(\begin{array}{rr} 36929 & 32 \\ 36928 & 33 \end{array}\right),\left(\begin{array}{rr} 12343 & 26906 \\ 17622 & 3851 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 36959 \end{array}\right),\left(\begin{array}{rr} 7712 & 10109 \\ 34507 & 26082 \end{array}\right)$.
The torsion field $K:=\Q(E[36960])$ is a degree-$313918488576000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/36960\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 127050fh
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210e1, its twist by $-55$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{385}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-55}) \) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{33})\) | \(\Z/16\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-55}, \sqrt{105})\) | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.275599841440000.59 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.275599841440000.1 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.1779622700625.4 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | 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 | \(\Z/24\Z\) | 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 | nonsplit | add | nonsplit | add | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
$\lambda$-invariant(s) | 5 | 1 | - | 1 | - | 1 | 1 | 1 | 1 | 3 | 1,1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 0 | 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.