Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-1070x-13207\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-1070xz^2-13207z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-86697x-9367839\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-22, 13\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.81765545757302127930088184419$ |
Torsion generators
\( \left(56, 325\right) \)
Integral points
\((-22,\pm 13)\), \((-19,\pm 25)\), \((56,\pm 325)\), \((212,\pm 3055)\)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 3380 \) | = | $2^{2} \cdot 5 \cdot 13^{2}$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $7140250000 $ | = | $2^{4} \cdot 5^{6} \cdot 13^{4} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{296747776}{15625} \) | = | $2^{8} \cdot 5^{-6} \cdot 13^{2} \cdot 19^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.64776022990016965374203547941\dots$ | ||
| Stable Faltings height: | $-0.43827194944032436141487104160\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.81765545757302127930088184419\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.83741358785033942320921397266\dots$ | ||
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: | $ 54 $ = $ 3\cdot( 2 \cdot 3 )\cdot3 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
| Torsion order: | $3$ | ||
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) $ ≈ $ 4.1082947421098084046191349538 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 1728 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
| $5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $3$ | $IV$ | Additive | 1 | 2 | 4 | 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 |
|---|---|---|
| $2$ | 2Cn | 2.2.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | split | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 3 | 2 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 3380.h
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.169.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.12338352.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $9$ | 9.3.95006081547000000.6 | \(\Z/9\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $18$ | 18.0.1878328153971890270208.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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.