Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-4963055x+5382131447\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-4963055xz^2+5382131447z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-79408875x+344377003750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-81, 76090\right)\) | \(\left(16819, 2154790\right)\) |
$\hat{h}(P)$ | ≈ | $0.18963354972936465952882715885$ | $1.2927256079087927870135536344$ |
Integral points
\( \left(-2447, 54796\right) \), \( \left(-2447, -52350\right) \), \( \left(-2381, 61990\right) \), \( \left(-2381, -59610\right) \), \( \left(-1485, 98086\right) \), \( \left(-1485, -96602\right) \), \( \left(-81, 76090\right) \), \( \left(-81, -76010\right) \), \( \left(595, 51078\right) \), \( \left(595, -51674\right) \), \( \left(819, 42790\right) \), \( \left(819, -43610\right) \), \( \left(1219, 33190\right) \), \( \left(1219, -34410\right) \), \( \left(1593, 38146\right) \), \( \left(1593, -39740\right) \), \( \left(2619, 100390\right) \), \( \left(2619, -103010\right) \), \( \left(3043, 134310\right) \), \( \left(3043, -137354\right) \), \( \left(6003, 435046\right) \), \( \left(6003, -441050\right) \), \( \left(16819, 2154790\right) \), \( \left(16819, -2171610\right) \), \( \left(35019, 6522790\right) \), \( \left(35019, -6557810\right) \)
Invariants
Conductor: | \( 76050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-4684151694643200000000 $ | = | $-1 \cdot 2^{18} \cdot 3^{6} \cdot 5^{8} \cdot 13^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{9836106385}{3407872} \) | = | $-1 \cdot 2^{-18} \cdot 5 \cdot 7^{3} \cdot 13^{-1} \cdot 179^{3}$ | 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: | $2.8695792099266669425525967509\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.035160221427556520905609143828\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.24503978680497446999568585006\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.12945955924200015876132688072\dots$ | 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: | $ 432 $ = $ ( 2 \cdot 3^{2} )\cdot2\cdot3\cdot2^{2} $ | 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: | $1$ | 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);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 13.704224888099094279067757743 $ | 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
$\displaystyle 13.704224888 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.129460 \cdot 0.245040 \cdot 432}{1^2} \approx 13.704224888$
Modular invariants
Modular form 76050.2.a.dg
For more coefficients, see the Downloads section to the right.
Modular degree: | 7257600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$5$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
$13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 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 |
---|---|---|
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 156 = 2^{2} \cdot 3 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 153 & 154 \\ 146 & 149 \end{array}\right),\left(\begin{array}{rr} 79 & 6 \\ 81 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 11 & 150 \\ 33 & 137 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 90 & 59 \\ 143 & 129 \end{array}\right),\left(\begin{array}{rr} 151 & 6 \\ 150 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[156])$ is a degree-$7547904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/156\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 76050.dg
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 650.f2, its twist by $-195$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.1300.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.87880000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.2088523125.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.593190000.2 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.773481589685361348519618896484375.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.37314516007272161853000000000000.1 | \(\Z/6\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | add | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 4 | - | - | 2 | 2 | - | 2 | 2 | 2 | 2 | 4 | 2 | 2,4 | 2 | 2 |
$\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.