Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-x^2+127x+38\)
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(homogenize, simplify) |
\(y^2z+yz^2=x^3-x^2z+127xz^2+38z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3+164160x+3756240\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(76, 665\right)\)
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$\hat{h}(P)$ | ≈ | $0.80184439404707595595484968129$ |
Integral points
\( \left(18, 88\right) \), \( \left(18, -89\right) \), \( \left(76, 665\right) \), \( \left(76, -666\right) \)
Invariants
Conductor: | \( 825 \) | = | $3 \cdot 5^{2} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-132867075 $ | = | $-1 \cdot 3 \cdot 5^{2} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{8990228480}{5314683} \) | = | $2^{18} \cdot 3^{-1} \cdot 5 \cdot 11^{-6} \cdot 19^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.24809533504155082665190056217\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.020144317030799235781559326701\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1590427469406408\dots$ | |||
Szpiro ratio: | $3.8922989829694825\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.80184439404707595595484968129\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.1249255113251152509022435129\dots$ | 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: | $ 2 $ = $ 1\cdot1\cdot2 $ | 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])
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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) $ ≈ $ 1.8040304299531682383269905694 $ | 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
$\displaystyle 1.804030430 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.124926 \cdot 0.801844 \cdot 2}{1^2} \approx 1.804030430$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 216 | 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 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
$5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
$11$ | $2$ | $I_{6}$ | nonsplit 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 |
---|---|---|
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 30.16.0-6.b.1.1, level \( 30 = 2 \cdot 3 \cdot 5 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 25 & 6 \\ 24 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 10 & 7 \end{array}\right),\left(\begin{array}{rr} 8 & 27 \\ 1 & 16 \end{array}\right)$.
The torsion field $K:=\Q(E[30])$ is a degree-$8640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 75 = 3 \cdot 5^{2} \) |
$3$ | nonsplit multiplicative | $4$ | \( 25 = 5^{2} \) |
$5$ | additive | $10$ | \( 33 = 3 \cdot 11 \) |
$11$ | nonsplit multiplicative | $12$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 825a
consists of 2 curves linked by isogenies of
degree 3.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.300.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.184528125.2 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.1350000.1 | \(\Z/6\Z\) | 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 |
$12$ | 12.0.1822500000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.59159048902684775903344659027099609375.4 | \(\Z/9\Z\) | not in database |
$18$ | 18.2.231627523606480125000000000000.2 | \(\Z/6\Z\) | 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 | ss | nonsplit | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 1,2 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 0,0 | 1 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.