Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+36x+144\) | (homogenize, simplify) |
\(y^2z=x^3+36xz^2+144z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+36x+144\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(6, 24\right)\) | \(\left(-2, 8\right)\) |
$\hat{h}(P)$ | ≈ | $0.34007526803572638204992425617$ | $0.71772583058596735289877375315$ |
Integral points
\((-3,\pm 3)\), \((-2,\pm 8)\), \((0,\pm 12)\), \((6,\pm 24)\), \((12,\pm 48)\), \((13,\pm 53)\), \((28,\pm 152)\), \((30,\pm 168)\), \((102,\pm 1032)\), \((126,\pm 1416)\), \((10176,\pm 1026516)\), \((25542,\pm 4082088)\)
Invariants
Conductor: | \( 5184 \) | = | $2^{6} \cdot 3^{4}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-11943936 $ | = | $-1 \cdot 2^{14} \cdot 3^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 432 \) | = | $2^{4} \cdot 3^{3}$ | 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: | $0.042014572893307436134481304785\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.3159632820940169372165787887\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7737056144690831\dots$ | |||
Szpiro ratio: | $2.7431147865896834\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.21613722830683635042282019694\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.5720623448583186721867710789\dots$ | 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: | $ 12 $ = $ 2^{2}\cdot3 $ | 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^{(2)}(E,1)/2! $ ≈ $ 4.0773743753186750769666825363 $ | 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
$\displaystyle 4.077374375 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.572062 \cdot 0.216137 \cdot 12}{1^2} \approx 4.077374375$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 1152 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 6 | 14 | 0 |
$3$ | $3$ | $IV$ | Additive | 1 | 4 | 6 | 0 |
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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$5$ | 5S4 | 5.5.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $160$, genus $4$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 66 \\ 78 & 49 \end{array}\right),\left(\begin{array}{rr} 88 & 69 \\ 27 & 103 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 60 & 61 \end{array}\right),\left(\begin{array}{rr} 59 & 15 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 81 & 100 \\ 10 & 81 \end{array}\right),\left(\begin{array}{rr} 109 & 42 \\ 66 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 60 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 59 & 60 \\ 30 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 97 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$221184$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 5184.d
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 324.b2, its twist by $24$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | 2.0.8.1-26244.5-a2 |
$3$ | 3.1.324.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.419904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.4478976.4 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.13436928.6 | \(\Z/6\Z\) | Not in database |
$12$ | 12.2.1624959306694656.3 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.20061226008576.4 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | 12.0.722204136308736.20 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.939901959762450051644918857728.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.84892385172761609433513984.1 | \(\Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | - | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-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.