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Properties

Label 10368.j1
Conductor $10368$
Discriminant $-186624$
j-invariant \( -3456 \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-18x+36\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-18xz^2+36z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-18x+36\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 0, 0, -18, 36])
 
Copy content gp:E = ellinit([0, 0, 0, -18, 36])
 
Copy content magma:E := EllipticCurve([0, 0, 0, -18, 36]);
 
Copy content oscar:E = elliptic_curve([0, 0, 0, -18, 36])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z\)

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(0, 6\right) \)$0.49382489396219784618445196198$$\infty$
\( \left(3, 3\right) \)$0.64707081869911222582450224875$$\infty$

$P$$\hat{h}(P)$Order
\([0:6:1]\)$0.49382489396219784618445196198$$\infty$
\([3:3:1]\)$0.64707081869911222582450224875$$\infty$

$P$$\hat{h}(P)$Order
\( \left(0, 6\right) \)$0.49382489396219784618445196198$$\infty$
\( \left(3, 3\right) \)$0.64707081869911222582450224875$$\infty$

Integral points

\((-5,\pm 1)\), \((-2,\pm 8)\), \((0,\pm 6)\), \((3,\pm 3)\), \((6,\pm 12)\), \((16,\pm 62)\), \((51,\pm 363)\) Copy content Toggle raw display

\([-5:\pm 1:1]\), \([-2:\pm 8:1]\), \([0:\pm 6:1]\), \([3:\pm 3:1]\), \([6:\pm 12:1]\), \([16:\pm 62:1]\), \([51:\pm 363:1]\) Copy content Toggle raw display

\((-5,\pm 1)\), \((-2,\pm 8)\), \((0,\pm 6)\), \((3,\pm 3)\), \((6,\pm 12)\), \((16,\pm 62)\), \((51,\pm 363)\) Copy content Toggle raw display

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: $N$  =  \( 10368 \) = $2^{7} \cdot 3^{4}$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Minimal Discriminant: $\Delta$  =  $-186624$ = $-1 \cdot 2^{8} \cdot 3^{6} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -3456 \) = $-1 \cdot 2^{7} \cdot 3^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $-0.27373896362432642926808641341$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $-1.2851432283316781479105304462$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.6131471927654584$
Szpiro ratio: $\sigma_{m}$ ≈ $2.23762823425727$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 2$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 2$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $0.29723026461907639654075633872$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $3.0282552117484675710864228294$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 6 $  = $ 2\cdot3 $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $1$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L^{(2)}(E,1)/2!$ ≈ $5.4005345875325654522667755272 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 5.400534588 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.028255 \cdot 0.297230 \cdot 6}{1^2} \\ & \approx 5.400534588\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([0, 0, 0, -18, 36]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([0, 0, 0, -18, 36]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   10368.2.a.j

\( q - q^{5} - 4 q^{7} - 2 q^{11} + q^{13} - 7 q^{17} - 8 q^{19} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 1152
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $III$ additive -1 7 8 0
$3$ $3$ $IV$ additive 1 4 6 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

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 $\ell$-adic index
$2$ 2G 4.2.0.1 $2$

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[1, 4, 0, 1], [13, 1, 2, 21], [17, 4, 10, 9], [2, 3, 19, 17], [1, 0, 4, 1], [21, 4, 20, 5], [13, 4, 2, 9], [1, 2, 2, 5]] GL(2,Integers(24)).subgroup(gens)
 
Copy content magma:Gens := [[1, 4, 0, 1], [13, 1, 2, 21], [17, 4, 10, 9], [2, 3, 19, 17], [1, 0, 4, 1], [21, 4, 20, 5], [13, 4, 2, 9], [1, 2, 2, 5]]; sub<GL(2,Integers(24))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.4.0-4.a.1.1, level \( 24 = 2^{3} \cdot 3 \), index $4$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 1 \\ 2 & 21 \end{array}\right),\left(\begin{array}{rr} 17 & 4 \\ 10 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 19 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 20 & 5 \end{array}\right),\left(\begin{array}{rr} 13 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[24])$ is a degree-$18432$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.

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
$2$ additive $2$ \( 81 = 3^{4} \)
$3$ additive $6$ \( 64 = 2^{6} \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 10368.j consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 10368.i1, 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
$3$ 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
$8$ 8.2.9172942848.3 \(\Z/3\Z\) not in database
$12$ 12.2.103997395628457984.16 \(\Z/4\Z\) 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 ord
$\lambda$-invariant(s) - - 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

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.