Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-80x-2400\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-80xz^2-2400z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6507x-1730106\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(15, 0\right) \)
Integral points
\( \left(15, 0\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 120 \) | = | $2^{3} \cdot 3 \cdot 5$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $-2400000000 $ | = | $-1 \cdot 2^{11} \cdot 3 \cdot 5^{8} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( -\frac{27995042}{1171875} \) | = | $-1 \cdot 2 \cdot 3^{-1} \cdot 5^{-8} \cdot 241^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.47999886506991459138871606774\dots$ | ||
| Stable Faltings height: | $-0.15538605044336860891041337693\dots$ | ||
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: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
| |||
| Real period: | $0.63474713898166649492320321353\dots$ | ||
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: | $ 8 $ = $ 1\cdot1\cdot2^{3} $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
| Torsion order: | $2$ | ||
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) $ ≈ $ 1.2694942779633329898464064271 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 64 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | Additive | -1 | 3 | 11 | 0 |
| $3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.48.0.175 |
The image of the adelic Galois representation has level $240$, index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 176 & 5 \\ 195 & 226 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 142 & 227 \end{array}\right),\left(\begin{array}{rr} 97 & 16 \\ 56 & 129 \end{array}\right),\left(\begin{array}{rr} 188 & 185 \\ 123 & 122 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 236 & 237 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 211 & 16 \\ 158 & 69 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 |
|---|---|---|---|
| Reduction type | add | split | split |
| $\lambda$-invariant(s) | - | 3 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 120.b
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | 2.2.24.1-600.1-p2 |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/8\Z\) | 2.0.4.1-1800.2-b1 |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $4$ | 4.0.55296.2 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | 4.2.55296.1 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.12230590464.4 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.1474560000.4 | \(\Z/16\Z\) | Not in database |
| $8$ | 8.2.113374080000.2 | \(\Z/6\Z\) | Not in database |
| $16$ | 16.0.88230205055486500383227904.149 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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.