Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-21420x+1206576\) | (homogenize, simplify) |
\(y^2z=x^3-21420xz^2+1206576z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-21420x+1206576\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(70, 224\right)\) |
$\hat{h}(P)$ | ≈ | $0.69190873220748374292040284330$ |
Torsion generators
\( \left(84, 0\right) \)
Integral points
\((-168,\pm 252)\), \((70,\pm 224)\), \( \left(84, 0\right) \), \((85,\pm 1)\), \((102,\pm 288)\)
Invariants
Conductor: | \( 28224 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $65548320768 $ | = | $2^{18} \cdot 3^{6} \cdot 7^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( 16581375 \) | = | $3^{3} \cdot 5^{3} \cdot 17^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[\sqrt{-7}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $1.1364621163983230474077751517\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.93904233603947808869203383481\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.1980441775334598\dots$ | |||
Szpiro ratio: | $4.052511320499709\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.69190873220748374292040284330\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $1.0441076544415988952242884345\dots$ | 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^{2}\cdot2^{2}\cdot2 $ | 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: | $2$ | 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) $ ≈ $ 5.7794176277825297904209902355 $ | 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
$\displaystyle 5.779417628 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.044108 \cdot 0.691909 \cdot 32}{2^2} \approx 5.779417628$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 32768 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
$3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | additive | $2$ | \( 63 = 3^{2} \cdot 7 \) |
$3$ | additive | $6$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
$7$ | additive | $20$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 28224.dw
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49.a3, its twist by $-24$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$4$ | 4.0.197568.2 | \(\Z/4\Z\) | not in database |
$6$ | 6.6.232339968.1 | \(\Z/14\Z\) | not in database |
$8$ | 8.0.624529833984.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.4.624529833984.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.156132458496.3 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.156132458496.28 | \(\Z/8\Z\) | not in database |
$8$ | 8.2.1053894094848.2 | \(\Z/6\Z\) | not in database |
$12$ | 12.12.3454839086735425536.2 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$20$ | 20.0.709769256018536283137282494313791488.2 | \(\Z/22\Z\) | 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 | add | add | ss | add | ord | ss | ss | ss | ord | ord | ss | ord | ss | ord | ss |
$\lambda$-invariant(s) | - | - | 1,1 | - | 1 | 1,1 | 1,1 | 1,1 | 1 | 1 | 3,1 | 1 | 1,1 | 1 | 1,1 |
$\mu$-invariant(s) | - | - | 0,0 | - | 0 | 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.