Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
\(y^2+xy+y=x^3-216x+2062\)
|
(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-216xz^2+2062z^3\)
|
(dehomogenize, simplify) |
\(y^2=x^3-279315x+97054254\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(20, 66\right) \)
Integral points
\( \left(2, 39\right) \), \( \left(2, -42\right) \), \( \left(20, 66\right) \), \( \left(20, -87\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
Conductor: | \( 102 \) | = | $2 \cdot 3 \cdot 17$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
Discriminant: | $-1228691592 $ | = | $-1 \cdot 2^{3} \cdot 3^{12} \cdot 17^{2} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
j-invariant: | \( -\frac{1107111813625}{1228691592} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-12} \cdot 5^{3} \cdot 17^{-2} \cdot 2069^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $0.43869376662733746355917130016\dots$ | ||
Stable Faltings height: | $0.43869376662733746355917130016\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
Real period: | $1.3930324921905334731674095343\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
| |||
Tamagawa product: | $ 24 $ = $ 1\cdot( 2^{2} \cdot 3 )\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
Torsion order: | $6$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
| |||
Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
Special value: | $ L(E,1) $ ≈ $ 0.92868832812702231544493968953 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
Modular degree: | 48 | ||
$ \Gamma_0(N) $-optimal: | no | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2B | 8.6.0.5 |
$3$ | 3B.1.1 | 3.8.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
$p$ | 2 | 3 | 17 |
---|---|---|---|
Reduction type | nonsplit | split | nonsplit |
$\lambda$-invariant(s) | 1 | 1 | 0 |
$\mu$-invariant(s) | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 102.b
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.8.1-5202.5-d5 |
$4$ | 4.2.9248.1 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.1212153856.5 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.5473632256.5 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$9$ | 9.3.1198435061547.1 | \(\Z/18\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.192769755062867795418284820529152.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.