Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-1218563x+516972656\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-1218563xz^2+516972656z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1579257675x+24143565111750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(750, 4687\right)\) | \(\left(1850, 66887\right)\) |
$\hat{h}(P)$ | ≈ | $1.3728535291502225129425497441$ | $4.9952737998463697362257750969$ |
Torsion generators
\( \left(-1275, 637\right) \), \( \left(625, -313\right) \)
Integral points
\( \left(-1275, 637\right) \), \( \left(264, 14488\right) \), \( \left(264, -14753\right) \), \( \left(530, 4247\right) \), \( \left(530, -4778\right) \), \( \left(625, -313\right) \), \( \left(661, 593\right) \), \( \left(661, -1255\right) \), \( \left(750, 4687\right) \), \( \left(750, -5438\right) \), \( \left(1100, 22012\right) \), \( \left(1100, -23113\right) \), \( \left(1850, 66887\right) \), \( \left(1850, -68738\right) \), \( \left(11569, 1233167\right) \), \( \left(11569, -1244737\right) \)
Invariants
Conductor: | \( 27075 \) | = | $3 \cdot 5^{2} \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $120573447359765625 $ | = | $3^{8} \cdot 5^{8} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{272223782641}{164025} \) | = | $3^{-8} \cdot 5^{-2} \cdot 6481^{3}$ | 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: | $2.2212344639973992845699129461\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.055703981802871132734980436459\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.038972011651739\dots$ | |||
Szpiro ratio: | $5.2568319877106235\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $6.8575048310480019863121788049\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.32754199291830234702375991848\dots$ | 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: | $ 32 $ = $ 2\cdot2^{2}\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)
|
Torsion order: | $4$ | 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^{(2)}(E,1)/2! $ ≈ $ 4.4922415976166975985896004582 $ | 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
$\displaystyle 4.492241598 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.327542 \cdot 6.857505 \cdot 32}{4^2} \approx 4.492241598$
Modular invariants
Modular form 27075.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 331776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$5$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
$19$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 8.48.0.123 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 4545 & 16 \\ 4544 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2889 & 2888 \\ 418 & 4295 \end{array}\right),\left(\begin{array}{rr} 1 & 1216 \\ 0 & 1141 \end{array}\right),\left(\begin{array}{rr} 3041 & 2888 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 4559 \end{array}\right),\left(\begin{array}{rr} 1139 & 1672 \\ 2470 & 2203 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4560])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 27075.f
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15.a2, its twist by $-95$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{95}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{10}, \sqrt{38})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.20851360000.1 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.521284000000.22 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.5337948160000.9 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.28493690558847385600000000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.7657353369870241665908736000000000000.22 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | 16.16.7657353369870241665908736000000000000.7 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | nonsplit | add | ss | ord | ord | ord | add | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 2 | - | 2,2 | 2 | 2 | 2 | - | 2,2 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.