Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-211x+224336\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-211xz^2+224336z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-272835x+10467450558\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(10, 467\right)\) | \(\left(388, 7460\right)\) |
$\hat{h}(P)$ | ≈ | $0.63815860454603260281646355238$ | $0.65843568332148194509300568975$ |
Integral points
\( \left(-62, 35\right) \), \( \left(-62, 26\right) \), \( \left(10, 467\right) \), \( \left(10, -478\right) \), \( \left(28, 476\right) \), \( \left(28, -505\right) \), \( \left(64, 656\right) \), \( \left(64, -721\right) \), \( \left(94, 971\right) \), \( \left(94, -1066\right) \), \( \left(388, 7460\right) \), \( \left(388, -7849\right) \), \( \left(2824, 148664\right) \), \( \left(2824, -151489\right) \), \( \left(1573894, 1973740859\right) \), \( \left(1573894, -1975314754\right) \)
Invariants
Conductor: | \( 92778 \) | = | $2 \cdot 3 \cdot 7 \cdot 47^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-21743960596218 $ | = | $-1 \cdot 2 \cdot 3^{15} \cdot 7^{3} \cdot 47^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{467103625}{9843350202} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-15} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3} \cdot 47$ | 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: | $1.2380963670454778110960715079\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.59640510009380137995924639627\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.080662648870203\dots$ | |||
Szpiro ratio: | $3.33670413982935\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.40391238155161842518302110971\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.54279413910134912495659507677\dots$ | 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: | $ 45 $ = $ 1\cdot( 3 \cdot 5 )\cdot3\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: | $1$ | 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! $ ≈ $ 9.8658573037508868112415135015 $ | 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
$\displaystyle 9.865857304 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.542794 \cdot 0.403912 \cdot 45}{1^2} \approx 9.865857304$
Modular invariants
Modular form 92778.2.a.g
For more coefficients, see the Downloads section to the right.
Modular degree: | 311040 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$47$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 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 |
---|---|---|
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7896 = 2^{3} \cdot 3 \cdot 7 \cdot 47 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 3949 & 6 \\ 3951 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 4278 & 3625 \\ 2635 & 6270 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 6219 & 2 \\ 7570 & 7 \end{array}\right),\left(\begin{array}{rr} 1975 & 6 \\ 5925 & 19 \end{array}\right),\left(\begin{array}{rr} 2257 & 6 \\ 6771 & 19 \end{array}\right),\left(\begin{array}{rr} 7891 & 6 \\ 7890 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[7896])$ is a degree-$44346337394688$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7896\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 92778.g
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-47}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.371112.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.23137651579392.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.99077043024.3 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.6473033477568.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.635836530893445786465982778742897791899392.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.1874680616046026914015585724200282943913984.2 | \(\Z/6\Z\) | 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 | split | ss | split | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | add |
$\lambda$-invariant(s) | 3 | 5 | 2,2 | 3 | 2 | 2 | 2 | 4 | 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 | - |
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.