Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-20439048x+35551551808\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-20439048xz^2+35551551808z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-26489006883x+1659090536253918\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{10683}{4}, \frac{27757}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $3.7256832805841443270793564274$ |
Integral points
None
Invariants
Conductor: | \( 67270 \) | = | $2 \cdot 5 \cdot 7 \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $192588298777000000000 $ | = | $2^{9} \cdot 5^{9} \cdot 7 \cdot 31^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1063985165884855369}{217000000000} \) | = | $2^{-9} \cdot 5^{-9} \cdot 7^{-1} \cdot 19^{3} \cdot 31^{-1} \cdot 53731^{3}$ | 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: | $2.8897351276502954509958389689\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.1727415254077223280312568066\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9570494107259142\dots$ | |||
Szpiro ratio: | $5.587428230336705\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.7256832805841443270793564274\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.17409177367005717443733783221\dots$ | 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: | $ 4 $ = $ 1\cdot1\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)
|
Torsion order: | $1$ | 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'(E,1) $ ≈ $ 2.5944432417990838937031205543 $ | 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
$\displaystyle 2.594443242 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.174092 \cdot 3.725683 \cdot 4}{1^2} \approx 2.594443242$
Modular invariants
Modular form 67270.2.a.i
For more coefficients, see the Downloads section to the right.
Modular degree: | 6220800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
$5$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$31$ | $4$ | $I_{1}^{*}$ | 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 | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 78120 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 19531 & 39078 \\ 0 & 2171 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 58591 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 62487 & 78112 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 39051 & 78112 \end{array}\right),\left(\begin{array}{rr} 78103 & 18 \\ 78102 & 19 \end{array}\right),\left(\begin{array}{rr} 44644 & 9 \\ 55791 & 78100 \end{array}\right),\left(\begin{array}{rr} 15116 & 78111 \\ 5049 & 20 \end{array}\right)$.
The torsion field $K:=\Q(E[78120])$ is a degree-$35829514764288000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/78120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 67270.i
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 2170.d1, its twist by $-31$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{93}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.3.8680.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.653972032000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.618647323959.2 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.1352981697498333.4 | \(\Z/9\Z\) | Not in database |
$6$ | 6.6.63061588800.1 | \(\Z/6\Z\) | 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 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.47520986095727334266328659795707047407616000000.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.18.497086299116159885607442651544721538999928872194048000000.1 | \(\Z/18\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | ord | nonsplit | split | ord | ord | ord | ord | ord | ord | add | ord | ss | ord | ord |
$\lambda$-invariant(s) | 9 | 3 | 1 | 4 | 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 | 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.