# Properties

 Label 960o5 Conductor $960$ Discriminant $1.393\times 10^{12}$ j-invariant $$\frac{35578826569}{5314410}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-4385x+94815$$ y^2=x^3+x^2-4385x+94815 (homogenize, simplify) $$y^2z=x^3+x^2z-4385xz^2+94815z^3$$ y^2z=x^3+x^2z-4385xz^2+94815z^3 (dehomogenize, simplify) $$y^2=x^3-355212x+70185744$$ y^2=x^3-355212x+70185744 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 1, 0, -4385, 94815])

gp: E = ellinit([0, 1, 0, -4385, 94815])

magma: E := EllipticCurve([0, 1, 0, -4385, 94815]);

oscar: E = elliptic_curve([0, 1, 0, -4385, 94815])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{4}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(103, 864\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(49, 0\right)$$, $$(103,\pm 864)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$960$$ = $2^{6} \cdot 3 \cdot 5$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $1393140695040$ = $2^{19} \cdot 3^{12} \cdot 5$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{35578826569}{5314410}$$ = $2^{-1} \cdot 3^{-12} \cdot 5^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 23^{3}$ 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: $1.0545806117222147987174678853\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.014859840882296834591619703113\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0339287621454463\dots$ Szpiro ratio: $5.354889040767453\dots$

## 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: $0.81910691718318947792851914759\dots$ 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: $48$  = $2^{2}\cdot( 2^{2} \cdot 3 )\cdot1$ 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: $4$ 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: $L(E,1)$ ≈ $2.4573207515495684337855574428$ 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

$\displaystyle 2.457320752 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.819107 \cdot 1.000000 \cdot 48}{4^2} \approx 2.457320752$

# 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

$$q + q^{3} + q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

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: 1536
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 not semistable. There are 3 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $4$ $I_{9}^{*}$ additive -1 6 19 1
$3$ $12$ $I_{12}$ split multiplicative -1 1 12 12
$5$ $1$ $I_{1}$ split multiplicative -1 1 1 1

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 4.12.0.7
$3$ 3B 3.4.0.1

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 = [[89, 96, 90, 119], [1, 12, 12, 25], [1, 24, 0, 1], [97, 24, 96, 25], [1, 0, 24, 1], [15, 106, 14, 11], [61, 24, 92, 29], [78, 41, 43, 28], [112, 3, 117, 34]]

GL(2,Integers(120)).subgroup(gens)

Gens := [[89, 96, 90, 119], [1, 12, 12, 25], [1, 24, 0, 1], [97, 24, 96, 25], [1, 0, 24, 1], [15, 106, 14, 11], [61, 24, 92, 29], [78, 41, 43, 28], [112, 3, 117, 34]];

sub<GL(2,Integers(120))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$120 = 2^{3} \cdot 3 \cdot 5$$, index $384$, genus $5$, and generators

$\left(\begin{array}{rr} 89 & 96 \\ 90 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right),\left(\begin{array}{rr} 61 & 24 \\ 92 & 29 \end{array}\right),\left(\begin{array}{rr} 78 & 41 \\ 43 & 28 \end{array}\right),\left(\begin{array}{rr} 112 & 3 \\ 117 & 34 \end{array}\right)$.

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

The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.

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 $4$ $$5$$
$3$ split multiplicative $4$ $$320 = 2^{6} \cdot 5$$
$5$ split multiplicative $6$ $$192 = 2^{6} \cdot 3$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 960o consists of 8 curves linked by isogenies of degrees dividing 12.

## Twists

The minimal quadratic twist of this elliptic curve is 30a4, its twist by $-8$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{10})$$ $$\Z/2\Z \oplus \Z/4\Z$$ not in database $2$ $$\Q(\sqrt{-2})$$ $$\Z/12\Z$$ 2.0.8.1-450.2-a4 $4$ 4.4.92160.2 $$\Z/8\Z$$ not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ $$\Z/2\Z \oplus \Z/12\Z$$ not in database $6$ 6.2.34560000.1 $$\Z/12\Z$$ not in database $8$ 8.0.262144000000.9 $$\Z/4\Z \oplus \Z/4\Z$$ not in database $8$ 8.8.849346560000.5 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.5184000000.12 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.8493465600.20 $$\Z/24\Z$$ not in database $12$ 12.0.1194393600000000.1 $$\Z/3\Z \oplus \Z/12\Z$$ not in database $12$ deg 12 $$\Z/2\Z \oplus \Z/12\Z$$ not in database $12$ deg 12 $$\Z/24\Z$$ not in database $16$ deg 16 $$\Z/16\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/12\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ not in database $18$ 18.0.172713999781822935859200000000.1 $$\Z/36\Z$$ 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) 2 3 5 add split split - 3 1 - 0 0

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

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 $0$.