# Properties

 Label 370d4 Conductor $370$ Discriminant $-256572640900$ j-invariant $$-\frac{16048965315233521}{256572640900}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-5255x-149075$$ y^2+xy=x^3-5255x-149075 (homogenize, simplify) $$y^2z+xyz=x^3-5255xz^2-149075z^3$$ y^2z+xyz=x^3-5255xz^2-149075z^3 (dehomogenize, simplify) $$y^2=x^3-6810507x-6934811706$$ y^2=x^3-6810507x-6934811706 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -5255, -149075])

gp: E = ellinit([1, 0, 0, -5255, -149075])

magma: E := EllipticCurve([1, 0, 0, -5255, -149075]);

oscar: E = EllipticCurve([1, 0, 0, -5255, -149075])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(\frac{335}{4}, -\frac{335}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$370$$ = $2 \cdot 5 \cdot 37$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-256572640900$ = $-1 \cdot 2^{2} \cdot 5^{2} \cdot 37^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{16048965315233521}{256572640900}$$ = $-1 \cdot 2^{-2} \cdot 5^{-2} \cdot 11^{3} \cdot 23^{3} \cdot 37^{-6} \cdot 997^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.99008134209992377619261716730\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.99008134209992377619261716730\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.979554104171432\dots$ Szpiro ratio: $6.314644864377942\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.28010026565549123651566483966\dots$ 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: $24$  = $2\cdot2\cdot( 2 \cdot 3 )$ 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: $L(E,1)$ ≈ $1.6806015939329474190939890379$ 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

$\displaystyle 1.680601594 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.280100 \cdot 1.000000 \cdot 24}{2^2} \approx 1.680601594$

# 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

$$q + q^{2} - 2 q^{3} + q^{4} + q^{5} - 2 q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})$$

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: 576
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$37$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.12.0.12
$3$ 3B.1.2 3.8.0.2

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)];

gens = [[3961, 6, 3132, 73], [13, 24, 3732, 3133], [4417, 24, 4416, 25], [2221, 24, 0, 1], [1, 24, 0, 1], [15, 16, 314, 335], [1481, 24, 3700, 1], [1111, 24, 2781, 145], [1797, 10, 3764, 101], [1, 0, 24, 1]]

GL(2,Integers(4440)).subgroup(gens)

Gens := [[3961, 6, 3132, 73], [13, 24, 3732, 3133], [4417, 24, 4416, 25], [2221, 24, 0, 1], [1, 24, 0, 1], [15, 16, 314, 335], [1481, 24, 3700, 1], [1111, 24, 2781, 145], [1797, 10, 3764, 101], [1, 0, 24, 1]];

sub<GL(2,Integers(4440))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37$$, index $384$, genus $9$, and generators

$\left(\begin{array}{rr} 3961 & 6 \\ 3132 & 73 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 3732 & 3133 \end{array}\right),\left(\begin{array}{rr} 4417 & 24 \\ 4416 & 25 \end{array}\right),\left(\begin{array}{rr} 2221 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1481 & 24 \\ 3700 & 1 \end{array}\right),\left(\begin{array}{rr} 1111 & 24 \\ 2781 & 145 \end{array}\right),\left(\begin{array}{rr} 1797 & 10 \\ 3764 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[4440])$ is a degree-$167931740160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4440\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 370d consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.4.1-68450.5-h3 $2$ $$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ 2.0.3.1-136900.2-d1 $3$ 3.1.300.1 $$\Z/6\Z$$ Not in database $4$ 4.2.136900.1 $$\Z/4\Z$$ Not in database $4$ 4.0.2960.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\zeta_{12})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.1440000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.299865760000.8 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.1518070410000.8 $$\Z/12\Z$$ Not in database $8$ 8.0.709689600.2 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ 12.0.18662400000000.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $12$ deg 12 $$\Z/12\Z$$ Not in database $12$ deg 12 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $18$ 18.0.5019894225952389997502894782634700000000.4 $$\Z/18\Z$$ 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

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 37 split ord split split 2 2 1 1 1 1 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.