Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-56528x-4044879\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-56528xz^2-4044879z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-73260315x-187618961610\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-81, 143\right)\) |
$\hat{h}(P)$ | ≈ | $4.6152894324366610122432837822$ |
Torsion generators
\( \left(309, 2693\right) \)
Integral points
\( \left(-81, 143\right) \), \( \left(-81, -63\right) \), \( \left(309, 2693\right) \), \( \left(309, -3003\right) \), \( \left(1021, 31173\right) \), \( \left(1021, -32195\right) \)
Invariants
Conductor: | \( 4450 \) | = | $2 \cdot 5^{2} \cdot 89$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $4574461500620800 $ | = | $2^{15} \cdot 5^{2} \cdot 89^{5} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{799052001908021545}{182978460024832} \) | = | $2^{-15} \cdot 5 \cdot 89^{-5} \cdot 421^{3} \cdot 1289^{3}$ | 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: | $1.7173641441364667179812624680\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.4491244920641166555478025791\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.994424017009493\dots$ | |||
Szpiro ratio: | $5.290189289018645\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $4.6152894324366610122432837822\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.31471885958592679840067811153\dots$ | 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: | $ 75 $ = $ ( 3 \cdot 5 )\cdot1\cdot5 $ | 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) $ ≈ $ 4.3575558805063359131492606430 $ | 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
$\displaystyle 4.357555881 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.314719 \cdot 4.615289 \cdot 75}{5^2} \approx 4.357555881$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 21600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$89$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
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 \( 3560 = 2^{3} \cdot 5 \cdot 89 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 2671 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1961 & 10 \\ 2685 & 51 \end{array}\right),\left(\begin{array}{rr} 3551 & 10 \\ 3550 & 11 \end{array}\right),\left(\begin{array}{rr} 2671 & 1790 \\ 0 & 1247 \end{array}\right),\left(\begin{array}{rr} 1781 & 10 \\ 1785 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3505 & 3441 \end{array}\right)$.
The torsion field $K:=\Q(E[3560])$ is a degree-$952772198400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 4450.j
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 |
---|---|---|---|
$3$ | 3.3.17800.1 | \(\Z/10\Z\) | Not in database |
$6$ | 6.6.225590080000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | deg 8 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.4656612873077392578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 89 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ord | add | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | 2 | 1 | - | 3 | 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 |
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.