Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-11682947706x-486046291388268\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-11682947706xz^2-486046291388268z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-15141100227003x-22676930347710350826\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(70033318448599980604531934/46019639408152852225, 584322049528247216641612560504210040672/312187001968381137830000554625)$ | $59.031341165585541501633694112$ | $\infty$ |
Integral points
None
Invariants
Conductor: | $N$ | = | \( 105222 \) | = | $2 \cdot 3 \cdot 13 \cdot 19 \cdot 71$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $-520483235310766404806064$ | = | $-1 \cdot 2^{4} \cdot 3 \cdot 13^{7} \cdot 19 \cdot 71^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( -\frac{176352252185278046480312672913502369}{520483235310766404806064} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-1} \cdot 13^{-7} \cdot 19^{-1} \cdot 71^{-7} \cdot 313^{3} \cdot 1791634153^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2047194845509181386167252398$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $4.2047194845509181386167252398$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0247801371035123$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.0182463564071265$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $59.031341165585541501633694112$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $\Omega$ | ≈ | $0.0072607494521914176963087158749$ | 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: | $\prod_{p}c_p$ | = | $ 28 $ = $ 2^{2}\cdot1\cdot1\cdot1\cdot7 $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ | 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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Special value: | $ L'(E,1)$ | ≈ | $12.001129784844197341485129873 $ | 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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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 12.001129785 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.007261 \cdot 59.031341 \cdot 28}{1^2} \approx 12.001129785$
Modular invariants
Modular form 105222.2.a.l
For more coefficients, see the Downloads section to the right.
Modular degree: | 87588480 | 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 at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
$3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
$13$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
$19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
$71$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
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 |
---|---|---|
$7$ | 7B.1.3 | 7.48.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 736554 = 2 \cdot 3 \cdot 7 \cdot 13 \cdot 19 \cdot 71 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 631340 \\ 736540 & 210407 \end{array}\right),\left(\begin{array}{rr} 560197 & 14 \\ 238609 & 99 \end{array}\right),\left(\begin{array}{rr} 620257 & 14 \\ 659029 & 99 \end{array}\right),\left(\begin{array}{rr} 245519 & 14 \\ 245525 & 99 \end{array}\right),\left(\begin{array}{rr} 736541 & 14 \\ 736540 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 566581 & 14 \\ 283297 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right)$.
The torsion field $K:=\Q(E[736554])$ is a degree-$488833763281403904000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/736554\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | split multiplicative | $4$ | \( 52611 = 3 \cdot 13 \cdot 19 \cdot 71 \) |
$3$ | split multiplicative | $4$ | \( 35074 = 2 \cdot 13 \cdot 19 \cdot 71 \) |
$7$ | good | $2$ | \( 114 = 2 \cdot 3 \cdot 19 \) |
$13$ | nonsplit multiplicative | $14$ | \( 8094 = 2 \cdot 3 \cdot 19 \cdot 71 \) |
$19$ | split multiplicative | $20$ | \( 5538 = 2 \cdot 3 \cdot 13 \cdot 71 \) |
$71$ | split multiplicative | $72$ | \( 1482 = 2 \cdot 3 \cdot 13 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 105222.l
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
---|---|---|---|
$3$ | 3.1.52611.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.145622898175131.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | not in database |
$7$ | 7.1.1807654339634125248.1 | \(\Z/7\Z\) | not in database |
$8$ | deg 8 | \(\Z/3\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$18$ | 18.0.100676924556812029305860858301439601681823.1 | \(\Z/14\Z\) | not in database |
$21$ | 21.1.192146698709342514033082782378321109757709747904944533004652782809412166483968.1 | \(\Z/14\Z\) | 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 | 71 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | ord | ord | nonsplit | ord | split | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | 15 | 2 | 1 | 51 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
$\mu$-invariant(s) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.