Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-2460x+45949\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-2460xz^2+45949z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3188187x+2191627638\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(23, 35\right)\) |
$\hat{h}(P)$ | ≈ | $2.1377785778889888602815159076$ |
Torsion generators
\( \left(29, -7\right) \)
Integral points
\( \left(-11, 273\right) \), \( \left(-11, -263\right) \), \( \left(13, 121\right) \), \( \left(13, -135\right) \), \( \left(23, 35\right) \), \( \left(23, -59\right) \), \( \left(29, -7\right) \), \( \left(29, -23\right) \), \( \left(269, 4217\right) \), \( \left(269, -4487\right) \)
Invariants
Conductor: | \( 862 \) | = | $2 \cdot 431$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-451936256 $ | = | $-1 \cdot 2^{20} \cdot 431 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{1646417855125441}{451936256} \) | = | $-1 \cdot 2^{-20} \cdot 431^{-1} \cdot 118081^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $0.64322969144724518789124081989\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.64322969144724518789124081989\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $2.1377785778889888602815159076\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $1.6300467988459672647337154497\dots$ | 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: | $ 20 $ = $ ( 2^{2} \cdot 5 )\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)
|
Torsion order: | $5$ | 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);
|
Special value: | $ L'(E,1) $ ≈ $ 2.7877433020235444698421074851 $ | 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
$\displaystyle 2.787743302 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.630047 \cdot 2.137779 \cdot 20}{5^2} \approx 2.787743302$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 640 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $20$ | $I_{20}$ | Split multiplicative | -1 | 1 | 20 | 20 |
$431$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
---|---|---|
$5$ | 5B.1.1 | 5.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4310 = 2 \cdot 5 \cdot 431 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 4301 & 10 \\ 4300 & 11 \end{array}\right),\left(\begin{array}{rr} 1766 & 5 \\ 4275 & 4306 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 2587 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 4255 & 4191 \end{array}\right)$.
The torsion field $K:=\Q(E[4310])$ is a degree-$2065614048000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4310\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 862e
consists of 2 curves linked by isogenies of
degree 5.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.431.1 | \(\Z/10\Z\) | Not in database |
$6$ | 6.0.80062991.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | 8.2.1207474162042032.3 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.43269949427712824607643788187116052817226593017578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 431 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.