# Properties

 Label 2304.h2 Conductor $2304$ Discriminant $373248$ j-invariant $$8000$$ CM yes ($$D=-8$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-30x+56$$ y^2=x^3-30x+56 (homogenize, simplify) $$y^2z=x^3-30xz^2+56z^3$$ y^2z=x^3-30xz^2+56z^3 (dehomogenize, simplify) $$y^2=x^3-30x+56$$ y^2=x^3-30x+56 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -30, 56])

gp: E = ellinit([0, 0, 0, -30, 56])

magma: E := EllipticCurve([0, 0, 0, -30, 56]);

oscar: E = EllipticCurve([0, 0, 0, -30, 56])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(2, 2\right)$$ (2, 2) $\hat{h}(P)$ ≈ $1.0564386879975241787084950898$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(4, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$(-5,\pm 9)$$, $$(2,\pm 2)$$, $$\left(4, 0\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$2304$$ = $2^{8} \cdot 3^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $373248$ = $2^{9} \cdot 3^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$8000$$ = $2^{6} \cdot 5^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-2}]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.20124009630581192876727104788\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.2704066260598257565278177574\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1.0564386879975241787084950898\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $2.9086064170891649431108606804\dots$ 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: $4$  = $2\cdot2$ 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: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $3.0727643471708570018195951299$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 3.072764347 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.908606 \cdot 1.056439 \cdot 4}{2^2} \approx 3.072764347$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); 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)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); 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

$$q - 6 q^{11} + 6 q^{17} + 2 q^{19} + O(q^{20})$$

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 192
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## 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$(j)_{-}$
$2$ $2$ $III$ Additive -1 8 9 0
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

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 $\ell$.

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 2304.h consists of 2 curves linked by isogenies of degree 2.

## Twists

The minimal quadratic twist of this elliptic curve is 256.a2, its twist by $12$.

## 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.4.18432.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.0.6144.2 $$\Z/6\Z$$ Not in database $4$ 4.2.18432.1 $$\Z/6\Z$$ Not in database $8$ 8.0.5435817984.7 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.339738624.9 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.150994944.2 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $8$ 8.4.1358954496.3 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $12$ 12.0.1521681143169024.4 $$\Z/18\Z$$ Not in database $16$ 16.0.118192468620711297024.15 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ 16.8.118192468620711297024.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.1846757322198614016.7 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $16$ 16.0.29548117155177824256.6 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ 16.8.29548117155177824256.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $20$ 20.0.5016449852600330836716407488512.1 $$\Z/22\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ss ss ord ss ord ord ss ss ss ss ord ord ss - - 1,1 1,3 1 1,1 1 1 1,1 1,1 1,1 1,1 1 1 1,1 - - 0,0 0,0 0 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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.