Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+72501x-23020166\) | (homogenize, simplify) |
\(y^2z=x^3+72501xz^2-23020166z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+72501x-23020166\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 24336 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-253318923646304256 $ | = | $-1 \cdot 2^{15} \cdot 3^{6} \cdot 13^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{1331}{8} \) | = | $2^{-3} \cdot 11^{3}$ | 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: | $2.0213499060350872779422141470\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $-1.1448154369550654292127561741\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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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: | $0.15577699475557231356620050846\dots$ | 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: | $ 4 $ = $ 2\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)
|
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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.62310797902228925426480203384 $ | 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
$\displaystyle 0.623107979 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.155777 \cdot 1.000000 \cdot 4}{1^2} \approx 0.623107979$
Modular invariants
Modular form 24336.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 224640 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{7}^{*}$ | Additive | -1 | 4 | 15 | 3 |
$3$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$13$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 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 |
---|---|---|
$3$ | 3Ns | 3.6.0.1 |
$5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 481 & 480 \\ 1080 & 481 \end{array}\right),\left(\begin{array}{rr} 521 & 0 \\ 0 & 521 \end{array}\right),\left(\begin{array}{rr} 1351 & 630 \\ 0 & 1351 \end{array}\right),\left(\begin{array}{rr} 1261 & 30 \\ 1290 & 1297 \end{array}\right),\left(\begin{array}{rr} 1 & 1248 \\ 240 & 1 \end{array}\right),\left(\begin{array}{rr} 333 & 163 \\ 1550 & 687 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 1169 & 1410 \\ 375 & 869 \end{array}\right),\left(\begin{array}{rr} 736 & 1095 \\ 1425 & 151 \end{array}\right),\left(\begin{array}{rr} 781 & 0 \\ 1500 & 781 \end{array}\right),\left(\begin{array}{rr} 1039 & 1410 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1081 & 0 \\ 0 & 241 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 361 & 1410 \\ 750 & 181 \end{array}\right),\left(\begin{array}{rr} 16 & 375 \\ 765 & 1306 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1080 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$1610219520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 24336ch
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 338e1, its twist by $156$.
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 |
---|---|---|---|
$3$ | 3.1.104.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.2.105456.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.0.316368.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.0.316368.2 | \(\Z/5\Z\) | Not in database |
$6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.0.100088711424.4 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$8$ | 8.0.100088711424.7 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/15\Z\) | Not in database |
$20$ | 20.4.24760121725657457541881856000000000000000.1 | \(\Z/5\Z\) | 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 | add | add | ord | ord | ss | add | ord | ord | ord | ss | ss | ord | ss | ord | ord |
$\lambda$-invariant(s) | - | - | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 |
$\mu$-invariant(s) | - | - | 0 | 0 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.