Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2+14093473x-2298056691\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z+14093473xz^2-2298056691z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+18265140333x-107492110083666\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{651}{4}, -\frac{651}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 134310 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-181450188260383318257960 $ | = | $-1 \cdot 2^{3} \cdot 3^{6} \cdot 5 \cdot 11^{6} \cdot 37^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{174751791402194852399}{102423900876336360} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-1} \cdot 37^{-8} \cdot 5590799^{3}$ | 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: | $3.1523402593987146005361917367\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.9533926229995293285052199477\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0675015979740146\dots$ | |||
Szpiro ratio: | $5.165799149315833\dots$ |
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: | $0.059568000002051770160524969024\dots$ | 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: | $ 64 $ = $ 1\cdot2\cdot1\cdot2^{2}\cdot2^{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: | $2$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.95308800003282832256839950438 $ | 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
$\displaystyle 0.953088000 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.059568 \cdot 1.000000 \cdot 64}{2^2} \approx 0.953088000$
Modular invariants
Modular form 134310.2.a.r
For more coefficients, see the Downloads section to the right.
Modular degree: | 17694720 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$37$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 97680 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 97676 & 97677 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5336 & 62161 \\ 7183 & 71050 \end{array}\right),\left(\begin{array}{rr} 52801 & 17776 \\ 40568 & 44529 \end{array}\right),\left(\begin{array}{rr} 81038 & 86581 \\ 87769 & 22210 \end{array}\right),\left(\begin{array}{rr} 62159 & 0 \\ 0 & 97679 \end{array}\right),\left(\begin{array}{rr} 94733 & 17776 \\ 73744 & 88485 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 97582 & 97667 \end{array}\right),\left(\begin{array}{rr} 77716 & 86581 \\ 66759 & 22210 \end{array}\right),\left(\begin{array}{rr} 97665 & 16 \\ 97664 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[97680])$ is a degree-$70934367043584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97680\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 134310.r
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1110.k6, its twist by $-11$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{11}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-110}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-10}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{11})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{11}, \sqrt{-30})\) | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.959512576000000.61 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.48575324160000.182 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | 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 | 11 | 37 |
---|---|---|---|---|---|
Reduction type | nonsplit | nonsplit | split | add | split |
$\lambda$-invariant(s) | 3 | 0 | 1 | - | 1 |
$\mu$-invariant(s) | 1 | 0 | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.