Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-110037833145x-14049416991328979\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-110037833145xz^2-14049416991328979z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1760605330323x-899164448050384978\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-191854, 95927\right) \), \( \left(-191182, 95591\right) \)
Integral points
\( \left(-191854, 95927\right) \), \( \left(-191182, 95591\right) \)
Invariants
Conductor: | \( 101430 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $787379148285132224390625000000 $ | = | $2^{6} \cdot 3^{10} \cdot 5^{12} \cdot 7^{8} \cdot 23^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1718043013877225552292911401729}{9180538178765625000000} \) | = | $2^{-6} \cdot 3^{-4} \cdot 5^{-12} \cdot 7^{-2} \cdot 23^{-6} \cdot 31^{3} \cdot 193^{3} \cdot 733^{3} \cdot 2731^{3}$ | 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: | $4.9314196148718550324146402526\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.4091583960101435341643412624\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0469264639341715\dots$ | |||
Szpiro ratio: | $7.624267065306599\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.0082892299310134780092477741207\dots$ | 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: | $ 384 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot( 2 \cdot 3 ) $ | 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: | $4$ | 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: | $9$ = $3^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.7904736650989112499975192101 $ | 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
$\displaystyle 1.790473665 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.008289 \cdot 1.000000 \cdot 384}{4^2} \approx 1.790473665$
Modular invariants
Modular form 101430.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 382205952 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$7$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$23$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1103 & 3852 \\ 2754 & 3791 \end{array}\right),\left(\begin{array}{rr} 1939 & 6 \\ 3858 & 3859 \end{array}\right),\left(\begin{array}{rr} 2575 & 3852 \\ 644 & 3863 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2905 & 6 \\ 3816 & 3823 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 3848 & 3857 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3853 & 12 \\ 3852 & 13 \end{array}\right),\left(\begin{array}{rr} 2857 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$103413252096$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 101430.s
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 4830.be2, its twist by $21$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-42}, \sqrt{46})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{161})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$6$ | 6.2.2977309629.4 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.222919710806016.82 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.13932481925376.7 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.796594176.2 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$18$ | 18.0.106729935266278574185713817605495872255347412109375.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 23 |
---|---|---|---|---|---|
Reduction type | nonsplit | add | nonsplit | add | split |
$\lambda$-invariant(s) | 7 | - | 0 | - | 1 |
$\mu$-invariant(s) | 0 | - | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.