Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-5255026468x+146625389186306\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-5255026468xz^2+146625389186306z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6810514301907x+6840974589419210094\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(41830, -12793\right)\) | \(\left(9340, 9912902\right)\) |
$\hat{h}(P)$ | ≈ | $0.89285450368704804504043512444$ | $4.4176383281667705481261778792$ |
Torsion generators
\( \left(\frac{167415}{4}, -\frac{167419}{8}\right) \)
Integral points
\( \left(9340, 9912902\right) \), \( \left(9340, -9922243\right) \), \( \left(33280, 2915582\right) \), \( \left(33280, -2948863\right) \), \( \left(41830, -12793\right) \), \( \left(41830, -29038\right) \), \( \left(42010, 34682\right) \), \( \left(42010, -76693\right) \), \( \left(42160, 87857\right) \), \( \left(42160, -130018\right) \), \( \left(1120336, 1182846929\right) \), \( \left(1120336, -1183967266\right) \)
Invariants
Conductor: | \( 140790 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $10458740686532413860000 $ | = | $2^{5} \cdot 3^{8} \cdot 5^{4} \cdot 13 \cdot 19^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{341135431944367622806895041}{222309381060000} \) | = | $2^{-5} \cdot 3^{-8} \cdot 5^{-4} \cdot 13^{-1} \cdot 19^{-4} \cdot 443^{3} \cdot 1577267^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $3.9773156107371753823768121320\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $2.5050961211539551523722984161\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0212595730829512\dots$ | |||
Szpiro ratio: | $6.643677063221685\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $3.9341015153601340846348469715\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.079186604057064888069651936886\dots$ | 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);
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Tamagawa product: | $ 128 $ = $ 1\cdot2^{3}\cdot2^{2}\cdot1\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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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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! $ ≈ $ 9.9689004485479013688177994684 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 9.968900449 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.079187 \cdot 3.934102 \cdot 128}{2^2} \approx 9.968900449$
Modular invariants
Modular form 140790.2.a.w
For more coefficients, see the Downloads section to the right.
Modular degree: | 117964800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
$3$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$13$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$19$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9880 = 2^{3} \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 6176 & 1243 \\ 8649 & 8678 \end{array}\right),\left(\begin{array}{rr} 9873 & 8 \\ 9872 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7608 & 3 \\ 2285 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6179 & 6176 \\ 1258 & 6181 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4159 & 9872 \\ 6756 & 9847 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9874 & 9875 \end{array}\right),\left(\begin{array}{rr} 1977 & 8 \\ 7908 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[9880])$ is a degree-$49562556825600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9880\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 140790.w
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 7410.n1, its twist by $-19$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{26}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{494}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{19}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{19}, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.56382077440000.9 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | split | split | ord | ord | nonsplit | ord | add | ss | ord | ord | ord | ord | ss | ord |
$\lambda$-invariant(s) | 3 | 3 | 3 | 2 | 4 | 2 | 2 | - | 2,2 | 2 | 2 | 2 | 2 | 2,2 | 2 |
$\mu$-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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.