Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+8124402x-11887136703\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+8124402xz^2-11887136703z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+10529224965x-554637837690090\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 47775 \) | = | $3 \cdot 5^{2} \cdot 7^{2} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-95370835511179250711325 $ | = | $-1 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{17} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{48412981936758748562855}{77853743274432041397} \) | = | $3^{-2} \cdot 5 \cdot 7 \cdot 13^{-17} \cdot 19^{3} \cdot 41^{3} \cdot 14303^{3}$ | 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: | $3.0944938779831080615044528905\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.5019358677348724482201008777\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.056323104404563911796967313729\dots$ | 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: | $ 34 $ = $ 2\cdot1\cdot1\cdot17 $ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.9149855497551730010968886668 $ | 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
$\displaystyle 1.914985550 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.056323 \cdot 1.000000 \cdot 34}{1^2} \approx 1.914985550$
Modular invariants
Modular form 47775.2.a.be
For more coefficients, see the Downloads section to the right.
Modular degree: | 3818880 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$13$ | $17$ | $I_{17}$ | Split multiplicative | -1 | 1 | 17 | 17 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 52.2.0.a.1, level \( 52 = 2^{2} \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 27 & 2 \\ 27 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 51 & 0 \end{array}\right),\left(\begin{array}{rr} 51 & 2 \\ 50 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[52])$ is a degree-$1257984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/52\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 47775.be consists of this curve only.
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 |
---|---|---|---|
$3$ | 3.1.63700.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.210999880000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | split | add | add | ss | split | ord | ord | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 2 | 1 | - | - | 0,0 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
$\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.
Additional information
The elliptic curves 47775.be1 and 3675.g1 are $17$-congruent, meaning that their 17-torsion subgroups are isomorphic as Galois modules. The isomorphism in this case is antisymplectic, and this is the only known pair of non-isogenous elliptic curves (up to simultaneous quadratic twist) which are antisymplectically 17-congruent.
There are no known examples of congruences modulo primes greater than 17.
See N. Billerey, "On some remarkable congruences between two eliptic curves" [arXiv:1605.09205]; T. Fisher, "On families of 7-congruent and 11-congruent elliptic curves" [MR:3356045, 10.1112/S1461157014000059]; N. Freitas and A. Kraus, "On symplectic isomorphisms of the $p$-torsion of elliptic curves", [arXiv:1607.01218]. See also T. Fisher, "On pairs of 17-congruent elliptic curves" [arXiv:2106.02033] for an example of a symplectic mod-17 congruence.