# Properties

 Label 2310o2 Conductor $2310$ Discriminant $6.718\times 10^{17}$ j-invariant $$\frac{19170300594578891358373921}{671785075055001600}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{4}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3+x^2-5575730x+5065104575$$ y^2+xy+y=x^3+x^2-5575730x+5065104575 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+x^2z-5575730xz^2+5065104575z^3$$ y^2z+xyz+yz^2=x^3+x^2z-5575730xz^2+5065104575z^3 (dehomogenize, simplify) $$y^2=x^3-7226146107x+236425911251094$$ y^2=x^3-7226146107x+236425911251094 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 1, -5575730, 5065104575])

gp: E = ellinit([1, 1, 1, -5575730, 5065104575])

magma: E := EllipticCurve([1, 1, 1, -5575730, 5065104575]);

oscar: E = EllipticCurve([1, 1, 1, -5575730, 5065104575])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{4}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(-2727, 1363\right)$$, $$\left(1593, 14323\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-2727, 1363\right)$$, $$\left(1145, 12979\right)$$, $$\left(1145, -14125\right)$$, $$\left(1369, -685\right)$$, $$\left(1593, 14323\right)$$, $$\left(1593, -15917\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$2310$$ = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $671785075055001600$ = $2^{20} \cdot 3^{6} \cdot 5^{2} \cdot 7^{4} \cdot 11^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{19170300594578891358373921}{671785075055001600}$$ = $2^{-20} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-4} \cdot 11^{-4} \cdot 267635041^{3}$ 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: $2.5108584014722441549876318328\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $2.5108584014722441549876318328\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## 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: $0.26843000563938978863981367056\dots$ 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: $640$  = $( 2^{2} \cdot 5 )\cdot2\cdot2\cdot2^{2}\cdot2$ 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: $8$ 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: $L(E,1)$ ≈ $2.6843000563938978863981367056$ 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

$\displaystyle 2.684300056 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.268430 \cdot 1.000000 \cdot 640}{8^2} \approx 2.684300056$

# 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("anayltic 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

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})$$

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: 92160
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 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $20$ $I_{20}$ Split multiplicative -1 1 20 20
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

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
$2$ 2Cs 8.48.0.25

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 = [[833, 8, 832, 9], [1, 0, 8, 1], [287, 2, 822, 835], [1, 8, 0, 1], [5, 4, 836, 837], [241, 8, 124, 33], [679, 2, 654, 835], [3, 638, 28, 285], [425, 212, 212, 421]]

GL(2,Integers(840)).subgroup(gens)

Gens := [[833, 8, 832, 9], [1, 0, 8, 1], [287, 2, 822, 835], [1, 8, 0, 1], [5, 4, 836, 837], [241, 8, 124, 33], [679, 2, 654, 835], [3, 638, 28, 285], [425, 212, 212, 421]];

sub<GL(2,Integers(840))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 287 & 2 \\ 822 & 835 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 836 & 837 \end{array}\right),\left(\begin{array}{rr} 241 & 8 \\ 124 & 33 \end{array}\right),\left(\begin{array}{rr} 679 & 2 \\ 654 & 835 \end{array}\right),\left(\begin{array}{rr} 3 & 638 \\ 28 & 285 \end{array}\right),\left(\begin{array}{rr} 425 & 212 \\ 212 & 421 \end{array}\right)$.

The torsion field $K:=\Q(E[840])$ is a degree-$645121$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.

## $p$-adic regulators

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

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 split nonsplit split split nonsplit 3 0 3 1 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 2310o consists of 6 curves linked by isogenies of degrees dividing 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(i, \sqrt{15})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{14}, \sqrt{30})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.42693156000000.39 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.768797006670000.2 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.63456228123711897600000000.12 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ 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.