Minimal Weierstrass equation
\(y^2=x^3+x^2-8248x+285396\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(44, 98\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.77937937044769313002346382928$ |
Torsion generators
\( \left(51, 0\right) \)
Integral points
\((44,\pm 98)\), \( \left(51, 0\right) \), \((55,\pm 34)\), \((100,\pm 686)\)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 784 \) | = | $2^{4} \cdot 7^{2}$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $47225249792 $ | = | $2^{13} \cdot 7^{8} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{128787625}{98} \) | = | $2^{-1} \cdot 5^{3} \cdot 7^{-2} \cdot 101^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.98059094684257672815544561109\dots$ | ||
| Stable Faltings height: | $-0.68551130824502523381446288209\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.77937937044769313002346382928\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $1.1233153024138364501605822856\dots$ | ||
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 $ = $ 2\cdot2^{2} $ | ||
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) $ ≈ $ 1.7509775464191117511001905895 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 768 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
| $7$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 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 | 8.6.0.6 |
| $3$ | 3B | 9.12.0.1 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ss | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1,3 | - | 1,1 | 1 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 784.b
consists of 6 curves linked by isogenies of
degrees dividing 18.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/6\Z\) | 2.2.28.1-14.1-b5 |
| $4$ | 4.0.392.1 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \times \Z/6\Z\) | 4.4.12544.1-98.1-n10 |
| $6$ | 6.0.116169984.1 | \(\Z/6\Z\) | Not in database |
| $6$ | \(\Q(\zeta_{28})^+\) | \(\Z/18\Z\) | 6.6.1075648.1-56.1-a5 |
| $8$ | 8.0.9834496.2 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.4.40282095616.3 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.9834496.1 | \(\Z/12\Z\) | Not in database |
| $12$ | 12.0.13495465182560256.3 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $12$ | 12.0.275417656786944.2 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $12$ | \(\Q(\zeta_{56})^+\) | \(\Z/2\Z \times \Z/18\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | 16.0.24759631762948096.2 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | 16.8.1622647227216566419456.1 | \(\Z/2\Z \times \Z/12\Z\) | 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.