Minimal Weierstrass equation
\(y^2+xy=x^3-x^2-2889x+46413\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-6, 255\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.65746346182309810864390496117$ |
Torsion generators
\( \left(18, -9\right) \)
Integral points
\( \left(-6, 255\right) \), \( \left(-6, -249\right) \), \( \left(18, -9\right) \), \( \left(43, -4\right) \), \( \left(43, -39\right) \), \( \left(99, 801\right) \), \( \left(99, -900\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 6930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $641640605760 $ | = | $2^{6} \cdot 3^{12} \cdot 5 \cdot 7^{3} \cdot 11 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
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| j-invariant: | \( \frac{3658671062929}{880165440} \) | = | $2^{-6} \cdot 3^{-6} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-1} \cdot 19^{3} \cdot 811^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.97671020090022536678476966831\dots$ | ||
| Stable Faltings height: | $0.42740405656617052108714704985\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.65746346182309810864390496117\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.85598740585941022926486151076\dots$ | ||
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 $ = $ 2\cdot2^{2}\cdot1\cdot3\cdot1 $ | ||
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) $ ≈ $ 3.3766826587998068612229752735 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
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| Modular degree: | 13824 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
| $5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 2.3.0.1 |
| $3$ | 3B.1.2 | 3.8.0.2 |
$p$-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 | nonsplit | add | split | split | split | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 4 | 2 | 2 | 3 | 1,1 | 1 | 1 | 3 | 3 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 6930q
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{385}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | Not in database |
| $3$ | 3.1.9075.1 | \(\Z/6\Z\) | Not in database |
| $4$ | 4.0.221760.6 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{385})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $6$ | 6.0.247066875.1 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $6$ | 6.2.1553638865625.2 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.0.7289334581760000.8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.49177497600.9 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $18$ | 18.0.26389997546398813188890030447333380800000000.3 | \(\Z/18\Z\) | 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.