Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-97109312x+368356070499\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-97109312xz^2+368356070499z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1553748987x+23573234762966\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(5637, 4041\right)\) |
$\hat{h}(P)$ | ≈ | $0.48343685019116047708097956127$ |
Torsion generators
\( \left(5693, -2847\right) \)
Integral points
\( \left(3677, 245121\right) \), \( \left(3677, -248799\right) \), \( \left(5637, 4041\right) \), \( \left(5637, -9679\right) \), \( \left(5693, -2847\right) \), \( \left(5757, 5921\right) \), \( \left(5757, -11679\right) \), \( \left(7457, 238821\right) \), \( \left(7457, -246279\right) \)
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: | $59386932244684800000 $ | = | $2^{16} \cdot 3^{7} \cdot 5^{5} \cdot 7^{8} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1180838681727016392361}{692428800000} \) | = | $2^{-16} \cdot 3^{-1} \cdot 5^{-5} \cdot 7^{-2} \cdot 23^{-1} \cdot 10569721^{3}$ | 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: | $3.1183545690620221552541131665\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.5960933502003106570038141763\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9919076307042978\dots$ | |||
Szpiro ratio: | $5.7939556687536555\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.48343685019116047708097956127\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.16270738918372694379844473846\dots$ | 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);
|
Tamagawa product: | $ 640 $ = $ 2^{4}\cdot2\cdot5\cdot2^{2}\cdot1 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 12.585399636769319537761442474 $ | 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
$\displaystyle 12.585399637 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.162707 \cdot 0.483437 \cdot 640}{2^2} \approx 12.585399637$
Modular invariants
Modular form 101430.2.a.fj
For more coefficients, see the Downloads section to the right.
Modular degree: | 11796480 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$3$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$7$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$23$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 19313 & 8 \\ 19312 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 16559 & 0 \\ 0 & 19319 \end{array}\right),\left(\begin{array}{rr} 12769 & 12768 \\ 8638 & 4495 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7939 & 7938 \\ 18298 & 14155 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19314 & 19315 \end{array}\right),\left(\begin{array}{rr} 13084 & 8281 \\ 16583 & 2766 \end{array}\right),\left(\begin{array}{rr} 916 & 11039 \\ 15617 & 16554 \end{array}\right),\left(\begin{array}{rr} 7736 & 5523 \\ 2765 & 8282 \end{array}\right)$.
The torsion field $K:=\Q(E[19320])$ is a degree-$397106888048640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 101430ey
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 4830q1, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{345}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{161}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{105}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{105}, \sqrt{161})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | add | split | add | ord | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | 2 | - | 1 | 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 |
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.