Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-38760x+23169600\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-38760xz^2+23169600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-50232987x+1081151556534\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-330, 390\right)\) |
$\hat{h}(P)$ | ≈ | $1.7566874765069524717153137778$ |
Torsion generators
\( \left(120, 4440\right) \)
Integral points
\( \left(-330, 390\right) \), \( \left(-330, -60\right) \), \( \left(-240, 4440\right) \), \( \left(-240, -4200\right) \), \( \left(-48, 5016\right) \), \( \left(-48, -4968\right) \), \( \left(120, 4440\right) \), \( \left(120, -4560\right) \), \( \left(300, 6060\right) \), \( \left(300, -6360\right) \), \( \left(320, 6440\right) \), \( \left(320, -6760\right) \), \( \left(720, 18840\right) \), \( \left(720, -19560\right) \), \( \left(2640, 134040\right) \), \( \left(2640, -136680\right) \), \( \left(3120, 172440\right) \), \( \left(3120, -175560\right) \), \( \left(409620, 261958440\right) \), \( \left(409620, -262368060\right) \)
Invariants
Conductor: | \( 19110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-228248616960000000 $ | = | $-1 \cdot 2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 7^{2} \cdot 13 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{131425499875625809}{4658135040000000} \) | = | $-1 \cdot 2^{-21} \cdot 3^{-7} \cdot 5^{-7} \cdot 7^{4} \cdot 13^{-1} \cdot 43^{3} \cdot 883^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.0110237351020420664273437908\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.6867053769261565155764516669\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.082463603375719\dots$ | |||
Szpiro ratio: | $4.8123641941407636\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.7566874765069524717153137778\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.26166034281635734480154365511\dots$ | comment: Real Period
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);
|
Tamagawa product: | $ 1029 $ = $ ( 3 \cdot 7 )\cdot7\cdot7\cdot1\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
Torsion order: | $7$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 9.6527643938042283349763971057 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 9.652764394 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.261660 \cdot 1.756687 \cdot 1029}{7^2} \approx 9.652764394$
Modular invariants
Modular form 19110.2.a.cz
For more coefficients, see the Downloads section to the right.
Modular degree: | 197568 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $21$ | $I_{21}$ | Split multiplicative | -1 | 1 | 21 | 21 |
$3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$5$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$13$ | $1$ | $I_{1}$ | Non-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 |
---|---|---|
$7$ | 7B.1.1 | 7.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 8191 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7288 & 7 \\ 7273 & 10914 \end{array}\right),\left(\begin{array}{rr} 10907 & 14 \\ 10906 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 5453 & 10914 \end{array}\right),\left(\begin{array}{rr} 4376 & 7 \\ 4361 & 10914 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 848 & 7 \\ 3353 & 10914 \end{array}\right),\left(\begin{array}{rr} 2731 & 5474 \\ 0 & 10531 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 19110.cz
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.76440.1 | \(\Z/14\Z\) | Not in database |
$6$ | 6.0.9115194816000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | Not in database |
$8$ | deg 8 | \(\Z/21\Z\) | Not in database |
$12$ | deg 12 | \(\Z/28\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | split | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 9 | 4 | 2 | - | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.