Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-91839x+10922186\) | (homogenize, simplify) |
\(y^2z=x^3-91839xz^2+10922186z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-91839x+10922186\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(242, 1694\right)\) | \(\left(\frac{121}{4}, \frac{22869}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $1.1890774959753625790834177096$ | $1.6913767380546661345045855014$ |
Integral points
\((-242,\pm 4356)\), \((190,\pm 576)\), \((242,\pm 1694)\), \((914,\pm 26278)\)
Invariants
Conductor: | \( 121968 \) | = | $2^{4} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1960221078579456 $ | = | $-1 \cdot 2^{8} \cdot 3^{6} \cdot 7^{2} \cdot 11^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{2141392}{49} \) | = | $-1 \cdot 2^{4} \cdot 7^{-2} \cdot 11 \cdot 23^{3}$ | 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: | $1.7219513160433181294040150938\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.88804979719628061861305799094\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.7347507160711815234488222723\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.46648439524678771238417796025\dots$ | 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: | $ 12 $ = $ 1\cdot2\cdot2\cdot3 $ | 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: | $1$ | 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! $ ≈ $ 9.7108096642847646081266841614 $ | 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
$\displaystyle 9.710809664 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.466484 \cdot 1.734751 \cdot 12}{1^2} \approx 9.710809664$
Modular invariants
Modular form 121968.2.a.h
For more coefficients, see the Downloads section to the right.
Modular degree: | 760320 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_0^{*}$ | Additive | -1 | 4 | 8 | 0 |
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $32$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 9 \\ 87 & 28 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 103 & 160 \\ 0 & 143 \end{array}\right),\left(\begin{array}{rr} 157 & 12 \\ 156 & 13 \end{array}\right),\left(\begin{array}{rr} 73 & 9 \\ 51 & 28 \end{array}\right),\left(\begin{array}{rr} 125 & 156 \\ 63 & 167 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 159 & 163 \\ 161 & 164 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 130 & 153 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$4644864$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 121968ew
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 3388c1, its twist by $-132$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.484.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.937024.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.80992606464.5 | \(\Z/3\Z\) | Not in database |
$6$ | 6.2.25299648.1 | \(\Z/6\Z\) | 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 |
$12$ | 12.0.640072188923904.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.6.3175139326380113995432933569287159808.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.531295486313950860719528916025344.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | nonsplit | add | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | 2 | 2 | - | 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 |
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.